This is to certify that the dissertation entitled The Effect of Experimental Nonidealities on the Results of Electrode Kinetics Experiments presented by Edward W. Schindler, Jr. has been accepted towards fulfillment of the requirements for Ph . D . degree in Chemis try ’M was , Nam WW Date August 311 1982 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 urn”? fifiggflj-‘Pr nab us. my ' L MSU LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. THE EFFECT OF EXPERIMENTAL NONIDEALITIES ON THE RESULTS OF ELECTRODE KINETICS EXPERIMENTS By Edward W. Schindler, Jr. A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1982 ABSTRACT THE EFFECT OF EXPERIMENTAL NONIDEALITIES ON THE RESULTS OF ELECTRODE KINETICS EXPERIMENTS By Edward W. Schindler, Jr. Experiments designed to measure fast electrochemical reaction rates cannot be performed exactly as they are described by the theory. Factors such as instrumental nonidealities and the electro- static interaction between the electrode and the charged reactant (diffuse-layer adsorption) have negligible effects under most condi- tions, but might exert a greater influence as the method is pushed to measure faster rates. It is of interest to determine to what ex- tent these factors affect the accuracy of the measured rate constants. The transient techniques which were studied include coulostatics, galvanostatic double pulse, and potential-step experiments. A parallel simulation method was used to generate transients showing the influence of weak reactant adsorption or an instrumental nonideality such as finite potentiostat risetime. These simulated transients were then subjected to data analysis procedures to determine the accuracy of the extracted rate constants. found rent is Edward W. Schindler, Jr. Both reactant adsorption and finite potentiostat risetime were found to affect the shape of normal pulse polarograms when the cur- rent is sampled at or below the 1 ms time scale. Peaks resembling d.c. polarographic maxima were found for both cases. The true limit- ing current was attained with adsorbed reactant, while finite rise- time polarograms exhibited a plateau region in which the current decreased toward the true limiting value for many hundreds of milli- volts beyond the wave. Heterogeneous rate constants were derived from these simulated data both by nonlinear regression on the chronoamperometric decay transients and by a pulse polarographic method. The accuracies of the values obtained by both methods were comparable, showing large errors (10 - 1002) for typical experimental conditions and large rate constants (>0.l cm/s). Small-perturbation methods were also studied in a similar manner, and the derived rate constants were found to be extremely sensitive to both instrumental and chemical nonidealities for fast reactions. The shapes of the nonideal transients showed no anomalous features. The conventional analysis for galvanostatic double pulse experiments was found to be superior to other analyses, yielding fairly accurate rate constants even under conditions such that no reasonable value could otherwise be derived. ACKNOWLEDGMENT The author would like to express his thanks to Professor Michael Weaver for offering helpful suggestions and advice throughout the course of this work. ii EDUCATIONAL GENEALOGY M. J. Weaver D. Inman J. 0. Bockris H. J. T. Ellingham A. J. Allmand W. H. Nernst W. Ostwald C. Schmidt J. Liebig J. L. Gay-Lussac C. L. Berthollet iii TABLE OF CONTENTS Chapter ‘ Page LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . xi LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . .xviii CHAPTER 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . 1 1.1. Phenomenological Electron-Transfer Kinetics O O O O O O O O I O O O O O O 0 O O O O O O 5 1.2. Experimental Techniques Used to Study Fast Elec trade metics O O O O I O O O O O O O O O 7 1.3. Instrumental Nonidealities. . . . . . . . . . . . . 11 1.4. Weak Reactant Adsorption. . . . . . . . . . . . . . 13 1.5. Summary of Dissertation . . . . . . . . . . . . . . 22 CHAPTER 2. METHODS . . . . . . . . . . . . . . . . . . . . . 24 2.1. Digital Simulation. . . . . . . . . . . . . . . . . 27 2.2. Accuracy of Digital Simulation. . . . . . . . . . . 34 2.3. Parallel Simulation Method. . . . . . . . . . . . . 36 2.4. Details of Simulation Programming . . . . . . . . . 41 2.5. Nonlinear Regression Analysis . . . . . . . . . . . 41 2.6. Error Function Evaluation . . . . . . . . . . . . . 45 CHAPTER 3. POTENTIAL-STEP EXPERIMENTS FOR THE STUDY OF IRREVERSIBLE ELECTRODE REACTIONS . . . . . . . . . . . . . . . . . . . . 48 3.1. Description of Experiment . . . . . . . . . . . . . 49 3.2. Analysis of Data from Potential— Step Experiments. . . . . . . . . . . . . . . . . . 51 iv mL Chapter 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. CHAPTER 4. 4.1. 4.2. 4.3. 404. 4.5. 4.6. 4.7. 4.8. CHAPTER 5. 5.1. 5.2. 5.3. Unique Aspects of Simulation. . . . . . . Effect of Finite Potential Risetime . . Effect of Weak Reactant Adsorption. . . . Effect of Coupled Risetime/Adsorption . . Effect of Nonidealities on Normal Pulse Polarography. . . . . . . . . . . Correlation of Model with Experimental Data. 0 O O O O O O O O O O O O O O O O O POTENTIAL-STEP EXPERIMENTS FOR THE STUDY OF QUASI-REVERSIBLE ELECTRODE MOTIONS O O O O O O O O O O O O O O 0 Description of Experiment . . . . . . . Conventional Data Analysis. . . . . . . . Unique Aspects of Simulation. . . . . . . Effect of Risetime on Small-Step Chronoamperometry . . . . . . . . . . . . Effect of Adsorption on Small-Step Chronoamperometry . . . . . . . . . . . . Comparison of the Effect of Risetime on Normal Pulse Polarography and Large- Step Chronoamperometry. . . . . . . . . . Comparison of the Effect of Adsorption on Normal Pulse Polarography and Large- Step Chronoamperometry. . . . . . . . . . Implications for the Use of NOrmal Pulse Polarography. . . . . . . . . . . . . . . COULOSTATICS EXPERIMENTS. . . . . . . . Description of Experiment . . . . . . . . Analysis of Data from Coulostatics Experiments . . . . . . . . . . . . . . . Unique Aspects of Simulation. . . . . . . Page 54 55 62 68 7O 79 85 86 87 88 89 93 99 103 108 114 115 118 119 Chapter 5.4. Time. 0 O O 505. CHAPTER 6. MENTS . . 6.1. 6.2. Experiments . 6.3. 6.4. Adsorption. 6.5. 6.6. Initial Investigations. Description of Experiment . Analysis of Data from G.D.P. gression Analysis . . . . . 6.7. Analysis. . 6.8. Methods . . CHAPTER 7. 7.1. 7.2. APPENDIX. LIST OF REFERENCES. . Unique Aspects of Simulation. A SAMPLE SIMULATION PROGRAM. Effect of Finite Charge Injection Alternative Analysis Procedures . Effect of Weak Reactant Adsorption. GALVANOSTATIC DOUBLE PULSE EXPERI- Shape of Deviations due to Reactant Effect of Adsorption in Conventional SUGGESTIONS FOR FURTHER RESEARCH. Effect of Adsorption in Nonlinear Re- Implications for the Use of Small-Step Extension to Other Adsorption Isotherms . Page 121 127 143 144 146 149 150 152 153 160 166 169 170 171 173 180 LIST OF TABLES Table Page 3.1 Error in Rate Constants Derived from Chronoamperometric Transients for Ir- reversible Reactions Due to Finite Po- tentiostat Risetime . . . . . . . . . . . . . . . 58 3.2 Comparison of Results of Standard and "Time-shift" Analyses of Chronoampero- metric Transients with Finite Risetime. . . . . . 61 3.3 Error in Rate Constants Derived from Chronoamperometric Transients Due to Weak Reactant Adsorption. . . . . . . . . . . . . 65 3.4 Error in Rate Constants Derived from Chronoamperometric Transients with Varying Amounts of Weak Reactant Ad- sorption. . . . . . . . . . . . . . . . . . . . . 67 3.5 Error in Rate Constants Derived from Chronoamperometric Transients Due to Both Finite Risetime and Weak Reactant Adsorption. . . . . . . . . . . . . . . . . . . . 69 3.6 Comparison of Results from Pulse Polar- ographic and Chronoamperometric Analyses of Transients with Finite Risetime. . . . . . . 73 vii Table Page 3.7 Comparison of Results from Pulse Polaro- graphic and Chronoamperometric Analyses of Transients with Weak Reactant Adsorp- tion. 0 C O C O O I O O O O O O C O O I O O O O O O 78 3+ 3.8 Rate Constants for Cr(aq) Reduction Derived by Pulse Polarographic and Chronoampero- metric Analyses . . . . . . . . . . . . . . . . . . 81 3.9 Results of Simulation Analysis of Crizé) Data. 0 C O O O O O O O O O O O O O O O O O O O O O 83 4.1 Error in Rate Constants Derived from Chronoamperometric Transients for Quasi- Reversible Reactions Due to Finite Rise- time. . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Error in Rate Constants Derived from Chronoamperometric Transients Due to weak Reactant Adsorption. . . . . . . . . . . . . . 95 4.3 Error in Rate Constants Derived from Chronoamperometric Transients Due to weak Reactant Adsorption (K.ox i Kred) . . . . . . . 97 4.4 Error in Rate Constant Derived from Chronoamperometric Transients Due to Weak Reactant Adsorption (Cox i C ) . . . . . . . 98 red 4.5 Comparison of Chronoamperometric and Pulse Polarographic Analysis of Transients with Finite Risetime. . . . . . . . . . . . . . . . 102 viii Table 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Page Comparison of Chronoamperometric and Pulse Polarographic Analyses of Tran- sients with Weak Reactant Adsorption. . . . . . . . 109 Error in Rate Constants Derived from Coulostatic Transients Due to Finite Injection Time. . . . . . . . . . . . . . . . . . . 124 Error in Double-Layer Capacitances Derived from Coulostatic Transients Due to Finite Injection Time. . . . . . . . . . . . 128 Error in Rate Constants Derived from Coulostatic Transients Due to Weak Reactant Adsorption . . . . . . . . . . . . . . . . 133 Effect of Varying Reactant Concentration on Error in Rate Constant Derived from Coulostatic Data with Weak Reactant Adsorption. . . . . . . . . . . . . . . . . . . . . 135 Effect of Varying Adsorption Coef- ficients on Error in Rate Constant Derived from Coulostatic Data . . . . . . . . . . . 136 Error in Double-Layer Capacitance Derived from Coulostatic Transients Due to Weak Reactant Adsorption . . . . . . . . . . 139 Effect of Varying Reactant Concentra— tion on Error in DoubleéLayer ix Table 5.8 6.1 6.2 6.3 6.4 6.5 6.6 Page Capacitance Derived from Coulostatic Data with Weak Reactant Adsorption. . . . . . . . . 140 Effect of Varying Adsorption Co- efficients on Error in Double-Layer Capacitance Derived from Coulostatic Data. . O O O O O . 0 . . O O O O O . O O . O O O O 141 Effect of Adjustable Experimental Parameters on Error in Rate Constant and Double-Layer Capacitance Derived from G.D.P. Transients with weak Reactant Adsorption. . . . . . . . . . . . . . . . . . . . . 154 Error in Rate Constants Derived from G.D.P. Transients Due to Weak Reac- tant Adsorption (K.ox - red). . . . . . . . . . . . 155 Error in Double-Layer Capacitance Derived from G.D.P. Transients Due to Weak Reactant Adsorption (Kox - Kred) . . . . . . . 156 Error in Rate Constant Derived from G.D.P. Transients Due to Weak Reactant Adsorption (K.ox i Kred) . . . . . . . . . . . . . . 159 Error in Rate Constants Derived from G.D.P. Transients Due to Weak Reactant Adsorption (Cox 4 Cred) . . . . . . . . . . . . . . 161 Results of Conventional Analysis of G.D.P. Data with Weak Reactant Adsorption. . . . . . . . . 165 Figure 1.1 1.2 2.1 2.2 2.3 2.4 Diffuse-layer adsorption coefficient LIST OF FIGURES Kad (calculated from Equation 1.11) :15. potential for a 3+ ion in 1 M RF at a mercury electrode . . . . . Diffuse-layer adsorption coefficient Kad (calculated from Equation 1.11) Izg. potential for a 3+ ion in 0.1 M KF at a mercury electrode . . . . . Summary of the process used to determine the effects of nonidealities upon the re- sults of electrode kinetics experiments . Basic elements of digital simulation of electrochemical experiments. . . Standard deviation of simulation lg. simulation parameter Ax1 for typical systems. Curve 1: conventional simulation. Curve 2: parallel simulation. . . . Standard deviation of simulation 3g, computation time for typical systems. Curve 1: Curve 2: conventional simulation. parallel simulation. . . . xi Page 0 18 . 19 26 . 29 . 39 . 40 Figure 2.5 2.6 3.1 3.2 3.3 Page Experimental process modified to reflect the parallel simulation method. . . . . . . 42 Functions of g(z) to obtain f(z) in four quadrants. . . . . . . . . . . . . . . . . . . 47 Chronoamperometric transients for irreversible reactions illustrating the effects of potentiostat risetime, with 5 kf-0.3cm/s,C-1mM,D-lx10- cm2/s, E - -500 mV. Curve 1: step ideal system. Curve 2: T = 20 us. Curve 3: T - 50 ps.. . . . . . . . . . . . . . . . 56 Chronoamperometric transients illustrat- ing the effects of weak reactant adsorpton, with parameters as in Figure 3.1, ex- cept kf - 0.1 cm/s. Curve 1: ideal . i -5 system. Curve 2. Kad - Kad - 2 x 10 ' i I -5 s cm. Curve 3. Kad 2 x 10 cm, Kad 0. Curve 4: K1 - 0,.K - 2 x 10-5 cm . . . . . . . . 63 ad ad Pulse polarograms illustrating the ef- fects of finite risetime, with a - 5 cmzls. 0.5,C-lmM,D-1x10- Curve 1: ideal system, 100 us sampling time. Curve 2: T - 20 us, 100 us sampling time. Curve 3: ideal system, 1 ms sampling time. Curve 4: T = 20 us, 1 ms sampling time . . . . . . . . . . . . . 71 xii Figure 3.4 3.5 3.6 4.1 Page Pulse polarograms illustrating the ef- fects of reactant adsorption, with 5 a- 0.5., c- 1mM, n- 1x 10' cm2/s. Curve 1: ideal system, 100 us sampling time. Curve 2: K1 a K - 2 x 10-5 ad ad cm, 100 us sampling time. Curve 3: ideal system, 1‘ms sampling time. i 5 Curve 4: Kad - Kad - 2 x 10 cm, 1 ms sampling time . . . . . . . . . . . . . . . . . . 75 Pulse polarograms illustrating the ef- fects of reactant adsorption, with a - 5 cmz/s, 100 0.5,C-lmM,D-1x10- us sampling time. Curve 1: ideal system. Curve 2: K: - 2 x 10.5 cm, d i Kad - 0. Curve 3. Kad - 0, Kad - 2 x 10-5 cm . . . . . . . . . O . O O O . O O O O 77 3+ Rate constants for Cr(OH2)6 reduc- tion derived from experimental chrono- amperometric transients using simula- tion analysis, Kg. potential. Line is least-squares fit of lower six points . . . . . . 82 Chronoamperometric transients for quasi-reversible reactions illustrating the effects of finite risetime, with kst d - 0.1 cm/s, a a 0.5, Estep - -10 mV, xiii ‘5. Figure 4.2 4.3 4.4 Page Cox - Cred ' 1 mM, Dox - Dred = 5 cm2/s. Curve 1: ideal system. 1 x 10- Curve 2: r - 50 us . . . . . . . . . . . . . . . . 90 Chronoamperometric transients il- 1ustrating the effects of reactant adsorption, with kstd - 0.1 cm/s, a ' 0.5, Estep I -10 mV, Cox = C 5 red = 1 mM, D - D - l x 10- cm2/s. ox red Curve 1: ideal system. Curve 2: 5 - K - 2 x 10- cm. . . . . . . . . . . . . 94 Kox red Pulse polarograms illustrating the ef- fects of finite risetime, with kStd = 0.1 cm/s, a - 0.5, Estd - 0 mV, cox - 5 1 mM, D - D - 1 x 10- cm2/s, ox re d 100 us sampling time. Curve 1: ideal system. Curve 2: T = 10 us. Curve 3: r = 20 us . . . . . . . . . . . . . . . . . . . 100 Pulse polarograms illustrating the ef- fects of reactant adsorption, with kstd - 0 mV, 5 d c - 1 mM, D - D - 1 x 10' ox ox red = 0.1 cm/s, a ' 0.5, E8t cm2/s, 100 us sampling time. Curve 1: ideal system. Curve 2: K1 a K - ox ox 2 x 10'5 cm. Curve 3: xix - 2 x 10‘5 1 cm, Kox 0. Curve 4. Kox - 0, KOx = 2 x 10..5 cm . . . . . . . . . . . . . . . . . . . . 105 xiv Figure 4.5 5.1 5.2 5.3 Page Pulse polarograms illustrating the effects of reactant adsorption at the onset of reversibility, with a = 0.5, E - 0 mV, C I 1 mM, D - D ox ox 5 2 i - on Is, KOx - KOx 2 x 10 std red . 1 x 10' 5 cm, 100 us sampling time. Curve 1: k - 3 cm/s. Curve 2: kst - 0.1 d m/s. . O . O O . . O . . . . . . I . O . O I O Q 107 std Coulostatic transients illustrating the effects of finite charge injection time, with k8t - 1 cm/s, C d d1 ' 20 uF/cmz. Q 2 cm ’ Cox Cred ' 1 x 10"5 inj - 1 nC, area - 0.02 - 1 mM, Dox - Dred cm2/s. Curve 1: ideal charge injection. Curve 2: tinj - 500 ns . O . O O . O . C O . . O O O O O O 0 O O O 122 Relative error in standard rate constant‘zg. ratio Tc/Td. Curve 1: t ITc - 0.0188. Curve 2: t inj/Tc - 0.133. inj - 0-0565. Curve 3: tinj/Tc Curve 4: tinj/Tc - 0.188. . . . . . . . . . . . . 126 Relative error in double-layer capaci- inj, Tc - 0.0188. Curve 2: tinj/Tc - 0.0565. tance 35. ratio Tc/Td. Curve 1: t Curve 3: t - 0.133. Curve 4: inj/Tc tinj/Tc - 0.188 . . . . . . . . . . . . . . . . . 129 XV Figure 5.4 6.1 6.2 Page Coulostatic transients illustrating the effects of reactant adsorption, with 2 k - 1 cm/s, Cd1 - 20 uF/cm , Qinj std - 1 nC, area - 0.02 cmz, C - C a ox red 1 mM, D - - D I 1 x 10-5 cm2/s. ox red Curve 1: ideal system. Curve 2: KOx - 1 x 10.6 cm. Curve 3: K a ox - 5 x 10"6 cm. . . . . . . . . . . . . . . . 131 ' K'red Kred Galvanostatic double pulse transients illustrating the effects of reactant adsorption, with kstd - 0.3 cm/s, c - 20 uF/cmz, 11 . 0.055 A/c.?, d1 2 . 1 Us, i2 - 0.002 A/cm , Cox 5 t1 C - 1 mM, D - D red ox red - 1 x 10 cmzls. Curve 1: ideal system. Curve 6 2: K - K - l x 10- cm. Curve 3: ox red d - 3 x 10-6 cm. Curve 4: K - K - 1 x 10-5 cm. 0 O O O O O O O O O O I 151 ox red Overpotential minimum gs, t? for con- ventional g.d.p. analysis, with kstd - KOX - Kre 0.3 cm/s, c - 20 uF/cmz, 1 - 0.003 2 - 1 mM’ Dox ' Dred 3 d1 2 A/cm ’ Cox 3 Cred 5 1 x 10- cm2/s. Lines are least- squares best fit. Curve 1: ideal system. Curve 2: Rex - K a 1 x 10.5 cm . . . . . . . . 163 red xvi Figure 6.3 Page Overpotential minimum gg_t? for con- ventional g.d.p. analysis, with kstd a 3 cm/s, 12 - 0.02 A/cmz, other param- eters as in Figure 6.2. Lines are least-squares best fit. Curve 1: ideal system. Curve 2: K0x - Kred - 5 1 x 10. cm . . . . . . . . . . . . . . . . . . . 164 xvii S 01 A c Cbulk O N red O X G. 2o 0 H red O std init step aim calc lim H NNH-HH'HHH-MWNNMNMUUUOHOHOOOOO LIST OF SYMBOLS Meanin electrode area concentration bulk concentration bulk concentration of species 0x bulk concentration of species Red concentration at electrode surface concentration at distance x from surface concentration at discrete point i in solution concentration at first discrete point in solution double-layer capacitance diffusion coefficient diffusion coefficient of species 0x diffusion coefficient of species Red electrode potential thermodynamic standard potential standard electrode potential electrode potential prior to.perturbation potential step size Faraday constant F/RT current simulated current calculated current polarographic limiting current size of first pulse in g.d.p. size of second pulse in g.d.p. general equilibrium constant xviii Usual Units cm mol/cm3 3 3 3 mol/cm mol/cm mol/cm mol/cm3 3 3 3 mol/cm mol/cm mol/cm uF/cm2 cm2/s 2 cm /s cmZ/s <<<< coul/mol uA uA uA uA A/ cm2 A/cm red std 5‘ 5‘ N‘ 8‘ H. HumH: ’Estep 5‘ overall - a .4. a5 62’ fil-D .0 .0 .C) N! U N bl N fl’ fi‘ fi' n Ii E- h) F‘ fi Q Meaning adsorption coefficient adsorption coefficient prior to perturbation adsorption coefficient of species 0x adsorption coefficient of species 0x prior to perturbation adsorption coefficient of species Red adsorption coefficient of species Red prior to perturbation standard heterogeneous rate constant forward rate constant forward rate constant for adsorbed species forward rate constant at potential Estep reverse rate constant effective rate constant in the presence of adsorption number of electrons size of current impulse in coulostatics amount of injected charge in coulostatics total charge on electrode (inside o.H.p.) charge on electrode charge of specifically adsorbed ions gas constant uncompensated resistance -temperature time length of first pulse in g.d.p. length of second pulse in g.d.p. time at which minimum appears in g.d.p. distance charge on diffuse-layer adsorbed ion charge on supporting electrolyte ion transfer coefficient dimensionless diffusion coefficient xix Usual Units cm cm cm cm cm cm cm/s cm/s cm/s cm/s cm/s uC uC/cm2 uC/cm2 uC/cmz J/mol'K us us us cm Meaning surface excess simulation time increment simulation distance increment distance increment between discrete points i-l and i first distance increment overpotential initial overpotential in coulostatics minimum overpotential in g.d.p. intercept of nmin vs. t? plot in g.d.p. standard deviation of simulation potentiostat rise time constant charge transfer time constant diffusion time constant flux of reactant at electrode surface faradaic flux diffuse-layer potential at o.H.p. diffuse-layer potential at distance x from o.H.p. Usual Units mol/cm2 s cm cm on mV mV mV mV us us us 2 mol/cm -s mol/cm -s mV mV CHAPTER 1 INTRODUCTION Reactions involving the transfer of electrons between two redox centers are fundamental in many fields, from biochemistry (photo- synthesis) to industrial processes (chlor-alkali technology). They are also the most basic type of reaction as there are no chemical bonds formed or broken. In many important systems, of course, the electron-transfer process is only one step in a mechanism involving several other, chemical steps, but it is generally the key step. Because these reactions are so basic, it is important to study their detailed mechanisms and rates so that the overall processes might be better understood. Electrochemical reactions (heterogeneous electron-transfer reac- tions) occur at the surface of an electrode immersed in a solution of an oxidizable or reducible (electroactive) species. In this special case of electron-transfer reactions, the electrode acts as the other reactant in the redox reaction, supplying or accepting electrons as required. The advantages of studying the kinetics of redox species in this manner are that it is relatively easy to follow the rate of the reaction, only the chemistry of a single species needs to be considered, and finally that the thermodynamics of the system (elec- trode plus reactant) are continuously variable by adjustment of the electrode potential. Various experimental techniques have been developed to study the kinetics of electrode reactions, and, like their chemical kinetics counterparts, they are subject to limitations based on the time scale of the method. Steady state methods are useful for studying rela- tively slow reactions, while transient techniques are used for rela- tively fast rates. It is important to know under what conditions the theory of a given experiment breaks down and the results yield no useful information. Electrochemical techniques, like all chemical experiments, are subject to minor experimental nonidealities. There are always factors which are not taken into account in the theory, but that do occur in the actual performance of an experiment. These effects must be minor, or the particular theory would not have general acceptance. This dissertation will examine a variety of electrochemical transient techniques to determine whether the nonidealities that are inevitable in actual experiments affect the results as the method is used to determine faster and faster reaction rates. Effects which are largely negligible when slow reactions are studied could induce significant error as the technique is taken to its limits. This is because the rate of a fast electrode reaction is controlled more by the diffusion of reactant molecules to the electrode surface than by the actual kinetics of the charge-transfer process. A given experie mental method must be sensitive enough to extract the small amount of kinetics information which is available in the data, and might therefore be more sensitive to nonideal experimental conditions. Both the instrumentation and the chemical system itself must lead to variations from the theoretical conditions for an experiment. Instruments rarely perform ideally; for example, a potentiostat re- quires a period of time to attain a new cell potential, even though a step-function is assumed in the derivation of the relevant equations. Additionally, coulombic attractive or repulsive forces between the reactant ions and the electrode are virtually universal, but seldom considered. Some previous work (1-4) has examined the effects of finite measure- ment precision and some instrumental limitations upon the results of many of the same electrode kinetics experiments which are examined here. These investigations simply used an explicit solution for the response of the system, rounded the results or otherwise modified the data to reflect the nonideal conditions, and derived the hetero- geneous rate constants from these modified data. Thus, no nonideali- ties could be treated in this manner in the absence of an explicit solution. The work described in this dissertation removes that restric- tion by using the digital simulation of electrochemical systems, which allows a much wider variety of nonideal conditions to be exr amined. Flanagan and Anson (5-7) examined some electrochemical systems to determine the effects of reactant adsorption for reversible systems. This work did not address the question of finite electron-transfer rates, but instead considered only deviations in the morphology response curves. They did, however, rely on digital simulation of the electrochemical experiments in some of their studies. The remainder of this chapter is devoted to some basic concepts of electron-transfer kinetics, a description of some transient tech- niques used to study fast electrode reactions, and a discussion of both the instrumental and chemical nonidealities which must occur, to at least some extent, in every experiment for the measurement of electrochemical reaction rates. The bulk of the dissertation examines the effects of these nonidealities on several common transient tech- niques and some general implications for the use of these methods. 1.1. Phenomenological Electron Transfer Kinetics For a chemically irreversible process -kf 0x + ne -* Red (1.1) occurring at an electrode surface, the rate constant kf is known to be a function of the electrode potential: anF kf . k td exp [- RT (E-Estdfl (1.2) s For these irreversible processes, kstd is simply the heterogeneous rate constant at some arbitrary potential Estd' This expression is in fact a linear free energy relationship, correlating the rate of the electron-transfer reaction to the free energy driving force as it varies with the electrode potential (AG - -nFE). If the process is chemically reversible, however, we must also consider the rate of the reverse reaction, -kf 0x + ne, Q? Red (1.3) The thermodynamic equilibrium of this redox process is governed by the Nernst equation (neglecting activity coefficients), (1.4) which specifies the potential at which no net reaction occurs for a given ratio of product to reactant concentration (the equilibrium potential). Since the Nernst equation defines the equilibrium constant at a given electrode potential, and this equilibrium constant must be equal to the ratio of forward and reverse rate constants (K - kf/kb), it is possible to derive an expression for the rate constant for the reverse reaction as a function of potential, kb =- kstd exp[S-]‘%1i.22-§ (E-E (1.5) std)] The standard potential, E for these chemically reversible std’ redox couples is now the thermodynamic E0; both the forward and reverse rate constants are equal to the standard rate constant at this potential. When this standard rate constant is very large (>10—100 cm/s), the couple is termed "reversible". At very small standard rate constants, the back reaction becomes insignificant (Eggg, in order to force kf to be large enough, the potential must be set sufficiently negative so that the reverse rate constant kb is very small). Under these conditions, the system behaves as if it were chemically irreversible, and is termed "irreversible". Redox couples with standard rate constants between these extremes are usually re- ferred to as "quasi-reversible". 1.2. Experimental.Techniques Used to Study Fast Electrode Kinetics Because of their speed, transient techniques (as opposed to steady-state methods) are usually used to study fast electrochemical reactions (9). These experiments involve, in the case of quasi- reversible reactions, a system at some equilibrium state to which a sudden perturbation is applied. The response of the system to the perturbation is followed, and rate parameters can be derived from these data. In the case of irreversible reactions, where no real redox equilibrium can exist, the perturbation usually involves a change in some experimental parameter which will initiate the electro- chemical reaction. The perturbations in either type of chemical system can he steps of potential, current, or charge, or continuously applied waveforms, such as an ac signal or a triangle wave. Potential-step experiments (10) are conceptually quite straight- forward. An electrode in a solution of the species of interest is held at a potential such that no reaction occurs and no current flows. The potential is rapidly changed to a new value, the reaction begins, land current flows. This current is monitored as a function of time. Various modes of this type of experiment are possible, depending on tihe chemistry of the reactant species. Large potential steps (sev- eral hundred millivolts) are needed if the reaction is irreversible (10), as one must start at a potential which yields a negligible rate c=01Msltant and step to one which causes the reaction to proceed at a faster rate. llarge potential steps can also be used for quasi-reversible re- actclc>ns (10) if the concentration of one half of the redox couple is very small. A large step from the equilibrium potential is required in order to change the relative concentrations of the oxidized and reduced forms of the reactant enough to yield an observable current. This mode is used in the normal pulse polarographic study of quasi- reversible redox reactions, and is characterized by the addition of only one form of the redox couple to the solution. Quasi-reversible reactions can also be studied when both the oxidized and reduced forms of the species are in solution at the start : of the experiment (11). The electrode is held at the equilibrium po— 1 tential for the system as given by the Nernst equation. A step in potential of several millivolts is applied, requiring the electrode reaction to proceed in order to adjust the concentration ratio of the reactants at the surface to that required by the Nernst equation. Analogous to the small potential-step experiments are coulostatics experiments (12,13) in which the perturbation of the system is a fast injection of charge into the cell. The charge then leaks off into the solution at a rate controlled by the electrode reaction rate and by the diffusion of fresh reactant to the surface. The overpotential is followed as it decays, since it is impossible to measure the current itself because the flow of electrons occurs only from the electrode surface to the solution once the charge has been injected. In small-step coulostatics for the study of quasi-reversible re- actions (12,13), both the oxidized and reduced species are in the solu- tion and the electrode is initially at the equilibrium potential. The equations describing the overpotential decay are somewhat more complicated, however, due to the fact that the potential (and hence the rate constants) is changing continuously throughout the experi- ment. The remaining step technique which has been commonly employed by electrode kineticists is galvanostatics (14). In these experiments, a system, again at such a potential that no current flows, is sub- jected to the sudden imposition of a constant current. The response of the system to this perturbation is a change in the overpotential sufficient to make the reactions proceed fast enough to consume these electrons as they flow through the cell. This overpotential-time transient contains information about the kinetics of the electron- transfer reaction. Galvanostatic double pulse experiments were developed (15) to deal with very fast electrode reactions by precharging the electrode double layer with a fast, high current pulse. The overpotential- time data are recorded immediately thereafter during the application of a smaller current to the cell. In addition to the step experiments, other types of transient signals have been used to measure electrochemical reaction rates. The application of an alternating potential waveform to a system at some equilibrium condition causes an alternating current to flow in response (16). Phase sensitive detection allows the extraction of heterogeneous rate data in these experiments. A potential ramp or triangular waveform can also be applied in techniques known as linear sweep voltammetry or cyclic voltammetry (17,18). A peak in the cur- rent is observed in the forward and reverse sweeps; the separation of these peaks can be related to the rate constant for the reaction of 10 interest. The responses of any of the transient techniques are controlled by the rate of charge-transfer and that of diffusion, the former dominating at short times and the latter at longer times. The faster the charge transfer process, the sooner the diffusion of reactants controls the response. These transient techniques are useful, there- fore, for the study of fast electrode reactions because the perturba- tion to the system can be made in a very small amount of time. Of the step methods, potentiostatics is the slowest because of its high demands on the instrumentation. At the time of the step, the potentiostat must provide a large amount of current to charge the double-layer capacitance, yet it must be sensitive enough to measure the small currents due to faradaic processes. Because of this and problems with uncompensated resistance, the potential-step method is not used on much less than a 0.1 ms time scale, which is suitable only for the measurement of heterogeneous rate constants up to about 0.1 cm/s. Charge-step methods are quite fast (indeed, this is the reason for their development),with the perturbation being applied in well under 1 us, depending on the solution resistance. Galvanostatic methods also can be employed on the microsecond time scale. In both these methods, the capacitance of the electrode has a direct effect on the resulting transient; there is no need for the instrumentation to "beat out" the charging process as in potentiostatics. These methods have been claimed to be suitable for the measurement of rate constants in the 0.1 to 10 cm/s range. 11 AC methods are limited by the frequency which can be applied to the cell (16). At higher frequencies, the technique is affected by stray capacitances and other irreproducible nonidealities which render the signal virtually useless for kinetics purposes. The upper limit of the accessible rate constant is roughly the same as for coulostatics and galvanostatics. Linear sweep methods are limited by the rate at which the potential can be scanned (17). At high sweep rates, the current which charges the double layer becomes quite large and deviations due to uncompensated resistance result, obscuring the kinetics information. Rate constants of about 0.1 cm/s are the maximum accessible with these methods. 1.3. Instrumental Nonidealities The instrumentation used to perform these transient techniques is basically the same as that required for steady-state methods, except that optimizations for a fast response time must be included. A typical potentiostat (19) contains a potential control amplifier whose function is to hold the working electrode at some fixed poten- tial relative to the reference electrode by supplying current to the cell. This current is measured by a second amplifier acting as a current to voltage converter. The response of an ideal potentiostat to an instantaneous step function in the control potential is a simultaneous step in the cell potential. In a real potentiostat, however, the response time is limited by the time constants of both the cell and the potentiostat itself, as wellas by the maximum current which can be supplied to 12 the auxillary electrode (20,22). Uncompensated resistance between the reference electrode and the working electrode is another factor which prevents the applied potenr tial from following the control potential (22). When current flows, the error in the cell potential is equal to iRu. Since the most cur- rent flows at the moment the step is applied, the potential only ap- proaches the control potential as the current decays. Although the amount of uncompensated resistance varies with the solution composition (as does the cell time constant), it is present to at least some extent in all experiments since it can never be per- fectly compensated. Because of this and the previously mentioned factors, all potential-step experiments must be to some degree non- ideal; an instantaneous potential step can never be applied to a real cell. Solution resistance is also a problem in performing coulostatics experiments. The theory (15) calls for an instantaneous injection of charge into the cell, but in practice a finite amount of time is needed to accomplish this. Additionally, current impulse charge in- jection experiments (23) have an inherent injection time during which a current pulse is applied. Galvanostatic experiments also suffer from iR drop errors in the measurement of the overpotential while current is applied. Not only is uncompensated resistance a problem, but the theory assumes that the current is initiated instantaneously (14). Again, instrumentation limitations prevent a perfect step function from being applied, lead- ing to deviations from the ideal predictions. 13 These types of nonidealities are of interest here because they are inevitable when experiments are performed with real instrumenta- tion. The nonidealities on which attention is focused in this work are finite potentiostat risetime and finite charge injection time in coulostatics. Uncompensated resistance effects were not considered because the effects in potentiostatics are similar to those of finite risetime. 1.4. Weak Reactant Adsorption In most electrochemical experiments, we are dealing with charged reactant ions in solution in the vicinity of‘a charged electrode. One would therefore expect some sort of electrostatic interaction to exist between them, either a concentration enhancement or reduction depending on the relative signs of the charges. A concentration en- hancement due to these factors can be looked upon as a form of ad- sorption, and can be referred to as diffuse-layer adsorption. This phenomenon is of interest because it must be present to at least some extent in almost all experiments which are designed to measure electrode kinetics (the exception being experiments which take place at a potential such that the total charge on the electrode is zero, EggL, the potential of zero charge, or p.z.c., in the absence of ionic specific adsorption). This electrostatic effect is well known in electrode kinetics through so-called double layer correc- tions of apparent heterogeneous rate constants, which consider the effect of the electrode charge on the relative stability of the transition state (24). 14 However, this correction does not consider the effect of the ex— cess number of molecules at the surface of the electrode upon the diffusion profile created during electrochemical processes. This extra reactant, known as the surface excess, will cause distortions in the diffusion profile because electrons which should be reacting with species diffusing from the bulk solution will be used in the re- action of the adsorbed species. The derivations of the equations des- cribing the transient responses assume the existance of an ideal, well-defined concentration gradient; hence, deviations from this ideal behavior will result. Because of the integrating effect of the dif- fusion profile, the transient will continue to be affected even after all the adsorbed species has reacted. It is of interest to determine the extent of the error in the rate constants which are derived from these distorted transients. In this study we will consider this adsorption to be relatively weak and to follow the Henry adsorption isotherm (25), I‘ a Ka ca (1.6) d This is the isotherm typically used for weak adsorption because it does not consider adsorbate-adsorbate interactions, and hence implies surface coverages low enough to ensure the absence of any such inter- acti.on effects. It will be shown below that the Henry isotherm is consistent with Gouy-Chapman double-layer theory (26) when the en- hancenmmt is not large. in: determine the extent of the electrostatic interactions, we 15 must consider the structure of the aqueous electrolyte solution in the vicinity of the charged electrode. A fixed layer of water mole- cules or specifically adsorbed anions is found at the surface of the electrode. This "inner layer" limits the access of other ions in the solution to the surface, in that the ions can progress only as far as some "plane of closest approach", or "outer Helmholtz plane" (o.H.p.). Gouy-Chapman theory defines the potential at that point in the inter- facial region, o2, with respect to the bulk solution, as a function of the total charge inside the o.H.p. (the actual charge on the elec- trode, qm, plus the charge of any adsorbed anions in the inner layer, q') to be 4,2 =- 21.52 s1nh’1[q/(11.74 015)] (1.7) for aqueous symmetrical electrolytes of charge 2 at 25 °C. Since this potential is relative to the bulk solution, it must decay to zero with increasing distance from the electrode. It is this region of potential decay that is called the "diffuse layer". The function describing the shape of this decay is also given by Gouy-Chapman theory: _ ¢ flzl 4),, ° T23]? tanh 1[tfimh(lea—flew(-I_ 0, C (2.1) ' Cbulk 29 m... J L Calculate Initial Boundary Conditions 1 Calculate Surface Boundary Conditions J Diffusion Over All Volume Elements Time - Time + At +—-—-‘ — ‘—l _l ‘Fl Figure 2.2. Basic elements of digital simulation of electro- chemical experiments. 30 is implemented by setting the concentration at each discrete point in the solution equal to the bulk value. If adsorption is present, the initial surface excess can also be calculated. The equations for the boundary conditions can become quite complex. In this work, four cases of surface boundary conditions are of interest: kinetically controlled chemically irreversible and reversible systems, both with and without adsorption. (If the reaction is fast enough, the response of the system is controlled solely by diffusion.) The irreversible systems will be used as examples. For the case in which no adsorption is present, the following equations must hold: 0 = D (BC/3x)x$0 (2.2) 0 = k C (2.3) The first is Fick's first law of diffusion, while the second is a statement of the rate law of the electrode reaction. Writing the equations in terms.of the discrete quantity Ax, o = D(C1-Co)/Ax (2.4) 0 = k Co (2.5) These equations can be solved simultaneously for the flux of re- actant at the surface and the concentration of reactant at the surface, - '1 .-. a +9.»... 31 o = kaID/(D+kax) (2.6) co = c1 - (FAX/D (2.7) The resulting expressions can be evaluated at each At time increment, using the appropriate rate constant if the potential is some func- tion of time. When weak adsorption is present, the equations become more complex. The adsorption process is assumed to be in equilibrium at all times and can be written for a species 0x as C (2.8) Fick's first law still holds, but the rate law is modified by allow- ing adsorbed species to react at the surface in addition to molecules diffusing from the bulk.solution, and by the addition of a term which allows for changing surface excesses: - k 00 + kaI‘ - AF/At (2.9) f f These equations can be solved for the parameters of interest, the flux ¢' Co’ and AF: 32 a co - rAx + AtDC1/[KadAx + AxAt(kf + Kadkf) + AtD] (2.10) a AP = AtD(C1-Co)/Ax - AtCo(kf + Kadkf) (2.11) a 0 = Co(kf + Kadkf) - AF/At (2.12) Now, however, the flux of electrons at the surface is no longer equal to the flux of molecules. To calculate the current, we con- sider only the faradaic flux, ofar = 00(1.f + Kadki) (2.13) This takes into account the possibility that the total flux includes some molecules coming from the bulk of solution to form or replenish the adsorbed layer. Of course, Kad can also be calculated as a func- tion of potential (and hence, time). Since there is a rapid equilibrium, the rate constants will always appear as a sum: 8 k a koverall f + Ka k (2°14) d f The following work has been done assuming that the rate constant for the adsorbed species was zero, in order that the rate constants kf might be equated with k overall for ease of interpretation. The existence of finite adsorption-desorption kinetics would complicate the calculation considerably; the rate constants would have to be " ”'1‘“? “‘5’ *t'fl I". r' 33 treated separately. For the quasi-reversible systems, we include terms for both the oxidized and reduced species and solve them in the same manner, although this is somewhat more complicated, especially with adsorbed reactant present. Once the concentration Co is calculated, the new concentrations at each point in the solution must be calculated. Fick's second law governs diffusion in the bulk solution, 2 2 3C/3t a D(3 C/ax ) (2.15) for planar geometry. This equation can be written in discrete terms for each point i as, 2 ACi a DAt(C1+1 - 2Ci + Ci-1)/(Ax) (2.16) In practice, the calculation is carried far enough out into the solu- tion that ACi is insignificant. After all the AC1 terms have been calculated for each species, the concentrations C are adjusted to i their new values, Ci = C1 + ACi (2.17) At this point, the time variable is incremented by the element At and the calculation of the boundary conditions begins with the new value of Cl (and possibly E, k, Kad’ or any other time-dependent term). 34 The computation continues in this manner, the flux being con- verted to current and recorded at the appropriate time interval until the calculation is complete. In this way it is possible to simulate a miriad of types of experiments by making relatively minor changes in the equations. It is this simplicity of modification and generality which makes digital simulation so valuable a tool in electrochemistry. 2.2. Accuragy of Digital Simulation Since digital simulation is a numerical solution to the relevant differential equations, it can provide only approximate results. The main factors which influence the accuracy of a given calculation are the simulation parameters Ax and At. Obviously, the finer the grid of time and distance increments, the closer the simulation ap- proaches the true solution. The choice of Ax and At is constrained by several factors. First, the parameter 8, defined as s = DAt/ (A102 (2.18) must always be less than 0.5 (27). Not meeting this condition will cause the ACi terms to be too large (larger than the corresponding C1), so that the concentration values may become negative and oscil- late. This is a mathematical constraint, general to all explicit finite difference and finite element solutions. A second constraint on the choice of the simulation parameters is practical in nature. As At or Ax approach zero, more and more 35 calculations must be performed, and many concentration values must be stored. Computer calculation speed is finite, so that a limita- tion is imposed on how fine a grid can be used. A related constraint is the precision (number of significant fig- ures) to which the calculation is carried. Smaller grid increments produce smaller changes and differences, so that accuracy will be limited by the precision of the computation. Of course, the pre- cision can be extended, but at a relatively large cost in computation time. Given these limitations, one can see that it is unreasonable to expect anything but an approximate solution to the problem. An accuracy of around one percent or less can be achieved in reasonable amounts of computation time, which makes simulations suitable for qualitative studies of the shapes of electrochemical transients, but limits their usefulness in more quantitative applications. An optimization of the procedure which was found to be success- ful involves the principle of expanding distance increments (31). Since one is interested mainly in conditions close to the surface, it makes sense to concentrate the calculation in this region. One does not need the same amount of precision far out in the solution, as the concentration gradient is quite small. Although the time in- crement size is limited by the smallest distance increment, Axl, the number of concentration points which need to be calculated at each time can be greatly reduced. An exponential expansion function was used for this work, Ax1 - Ax exp(const ° 1) (2-19) 1 36 The value of the constant can be varied; a value of 0.1 tended to limit most of the simulations to about twenty to thirty concentra~ tion values.. It is necessary to modify the discrete form of Fick's second law to reflect this unequal spacing, (c -C ) (C -c ) 1+1 1 1 1-1 ACi = 21311: ( “1+1 - Axi )/(Ax1+1 + Axi) (2.20) Of course, the equations for the boundary conditions use the value of Axl. The use of this method of expanding distance increments yields a computation time savings of about 50%, even though the diffusion equation is more complex. No adverse effect on the overall accuracy of the simulations from the use of this equation were observed unless the concentration grid was severely expanded. 2.3. Parallel Simulation Method (32) As the diffusion limit of a particular method is approached with larger and larger reaction rates, the shape of the transient depends less and less on the exact value of the rate constant. The rate constant which can then be derived from the transient depends heavily on minor variations in its shape. It is for this reason that some minor deviations from ideality produce no substantial error at slower rates, but induce significant error for faster reactions. Unfortunately, simulation inaccuracy has a similarly large effect on this parameter, sometimes even masking the deviations due to the nonideality itself. 37 It was found to be necessary to develop a scheme which would eliminate this error, recognizing that conventional methods of optimization were impractical due to computer time limitations. The simulated systems in this work deal with conditions which cause deviations from some ideal experiment. These deviations are generally minor, and a closed form solution is generally available for the exact case. The method developed to eliminate simulation inaccuracy uses parallel simulations of the nonideal system of interest and the cor- responding ideal experiment. All simulation parameters (Ax, At, etc.) are identical; the only difference between the resulting transients is due to the effect of the nonideality. The result of the parallel simulation is-a sort of error curve which gives the deviation from ideality at each point along the transient. This error curve can then be impressed on the exact ideal transient calculated from the closed form solution to yield a calculated, nonideal transient which reflects only the effect of the nonideality and shows no error from simulation inaccuracy. The process can be summarized by the followb ing equation: nonideal nonideal g ideal X (2.21) calculated calculated ( xideal )simulated where X is the measured quantity, usually a function of time. To verify that this scheme minimizes the dependence of the transients on the simulation parameters, a nonideal system was needed to which an exact solution has been derived. The case of a 38 linearly rising potential-step experiment (33) was chosen to serve as the nonideal case, while the well-known instantaneous-step experi- ment (34) was used as the ideal system. Simulations were performed as outlined above, and the results compared to the exact transient calculated from the equation through the use of a standard deviation from simulation, 1 2!: Osim . l:n 2(isim,- icalc) ] (2°22) Figure 2.3 shows the results for a typical set of parameters which shows the error, in arbitrary units, as a function of the simu- Osim’ lation parameter Ax1 for the standard simulation and the newly de- veloped parallel simulation scheme. It is clear that the new method produces simulations which are more accurate and less dependent on the actual simulation parameters than does the conventional method. This point is further illustrated in Figure 2.4, which displays the error as a function of computation time. Even though the parallel scheme requires twice the time for a given set of simulation param- eters, a great savings is gained in the time required for an accurate calculation. This parallel simulation scheme was used throughout this work when the resulting transient was to be subjected to nonlinear regres- sion analysis. Other analysis methods seem to be less sensitive to vari- ations in the shape of the transient; conventionally simulated tran- sients were used in these cases for convenience. The overall scheme of this investigation as shown in Figure 2.1 39 031m Figure 2.3. Standard deviation of simulation vs. simulation parameter Ax1 for typical systems. Curve 1: conventional simulation. Curve 2: parallel simulation. 40 sim Relative Computation Time Figure 2.4. Standard deviation of simulation vs. computation time for typical systems. Curve 1: conventional simulation. Curve 2: parallel simulation. 41 can now be modified to include the parallel simulations and the impres- sion of the error curve on the calculated ideal transient. The over- all process, as modified, is shown in Figure 2.5. 2.4. Details of Simulation Programming An example of the complete program used to generate the error curve in the above example is presented in the Appendix to illustrate the application of the concepts developed above. The language used in this work was FLECS (35), a structured pre-processor for FORTRAN. Translation of this code to FORTRAN would be a trivial matter for anyone with a knowledge of programming. Structured programming (36) was chosen for ease of coding, maintenance, and modification, and for the clarity of the final result. The programs were executed in an interactive environment on Digital Equipment Corporation LSI-ll and LSI-11/23 processors (37) under version 4 of operating system RT-ll (37). 2.5. NOnlinear Regression Analysis After the nonideal system is simulated, the apparent rate constants must be extracted from the transients. Although there are several ways of accomplishing this, nonlinear regression is a generally ap- plicable method which has been used extensively in this work. A brief outline of the technique and its use follows. There are many examples in electrochemistry and in chemistry in general of the analysis of experimental data by linearizing the 42 Known Rate Constant it Digital Simulation Nonideal Ideal System System I l Simul ted Transient Res onses Impress Errors on Calculated Ideal Transient Calculated Transient Response Nonlinear Regression Analysis 1! Apparent Rate Constant Figure 2.5. Experimental process modified to reflect the parallel simulation method. 43 dependence of some measured quantity on an independent variable. This linearized data can then be analyzed with graph paper and a ruler, or with the more sophisticated linear least squares calculation. This type of manipulation of the data is not always statistically sound, as when the same variable ends up plotted on both axes of the graph, and can lead to erroneous results (38). Additionally, approxi- mations of certain functions often must be made so that the equations may be reduced to a linear form. When these approximations do not hold, the line displays curvature, rendering any calculated slope and intercept meaningless. Even worse is the treatment of such curved plots with polynomial least squares; the resulting virial co- efficients have no physical significance whatsoever in most cases, and the resulting equation cannot be used for any sort of reliable extrapolation. Nonlinear regression (39) is a numerical method which can be used to fit an arbitrary equation to a set of data by adjusting certain key parameters. Many algorithms have been used for the actual adjust- ment of the parameters, but all attempt to minimize the overall resid- ual. At the minimum residual, the values of the adjustable paramr eters are those which make the equation best fit the data. One is able to use the theoretically predicted equation to fit the data as they are measured so as to avoid doubtful approximations in the analysis procedure. Of course, if the data are not actually described by the equation, the results of this method must be in doubt. It is possible to get an indication of the goodness of fit ‘by examining the individual residuals along the curve for evidence 44 of a systematic pattern (39). A perfect fit has all zero residuals, but we usually see a random scatter around zero due to the random errors of measurement. For a given fit to discriminate among correct and incorrect models, the systematic deviations must be large enough to be seen through the random scatter. Standard devia- tions from regression can also be calculated and compared to get an indication of which is the better fit. In this work, we are purposely fitting nonideal, simulated data to ideal models. In most of the cases, the systematic deviations would not be clearly evident were the transients recorded at normal measurement precision; however, the shape would differ enough that errors in the adjusted parameters would still result. Since the equations do not fit the data, a question arises about the interval over which data is to be analyzed. Using different ranges of data along the same nonideal transient will yield a variation in the results. (The same will be true for analyzing a curved line with linear regression.) For the following work, if an optimum time range for analyzing transients from which to derive rate parameters has been published, that range was used in these analyses. If no optimum range has been specified, a range consistent with a reasonable ex- perimentally accessable measurement time has been used. The nonlinear regression itself was done by program CFT4A (40). This routine, while not especially efficient, was developed specific- ally for small computer systems and has been applied successfully for 'many chemical systems and experiments. No changes were made to the calculational portion of the program, although some I/O and control parameters were added for convenience. CFT4A provides for a separate 45 subroutine which calculates (by any method) a set of data according to user-provided equations, the set of independent variables, and the current values of the adjustable parameters. This calculated data set is then compared to the experimental set. The equations in this routine can be simple algebraic equations, a complex integra- tion routine, or even a digital simulation procedure. 2.6. Error Function Evaluation The calculation of the ideal transients often requires the evalua- tion of the exponential error function, f(x) = exp(xz) erfc(x) (2.23) When x is a real number, a rational function approximation developed by Flanagan (41) was used, which is valid over a wide range of arguments. In equations describing the ideal transient for some electrochem- ical methods, however, the argument of the function is a complex number. A procedure has been published (42,43) for the evaluation of the so- called complex error function, g(z) = exp(-zz) erfc(iz), z = x + iy (2.24) which is valid for arguments z in the first quadrant (x and y greater than zero). The relationship between the function of interest, f(z), and the calculable function g(z) is trivial: 46 f(z) = g(iz) (2.25) It is also necessary to extend the evaluation of g(z) to other quadrants by means of the following relationships: g(-Z) = 2 exp (-22) - 3(2) (2.26) g(conj z) = conj g(—z) (2.27) conj z = x - iy (2.28) These equations allow the function f(z) to be calculated from variations of the function g(z). Figure 2.6 displays the functions which must be evaluated to obtain f(z) in the four quadrants. 47 Figure 2.6. Functions of g(z) to obtain f(z) in four quadrants. sigg of x sigg of 2 function to evaluate to obtain f(z) + + conj g(-iz) + - g(iz) - + 2exp{-(iz)z} - g<-iz) - - conj [2exp{-(conj iz)2}-g(conj iz)] z - x + iy i - JrI conj z - x - iy f(z) - exp (22) erfc (z) g(z) - exp (-zz) erfc (iz) CHAPTER 3 POTENTIALPSTEP EXPERIMENTS FOR THE STUDY OF IRREVERSIBLE ELECTRODE REACTIONS 48 3.1. Description of Experiment A species which undergoes an irreversible electron-transfer re- action can be conveniently studied with potential-step experiments (44). In these methods, the electrode is in a solution of the species under study at a potential such that no reaction occurs. The potential is then stepped and held constant at the new value. The electron transfer reaction starts at a rate determined by the rate constant kf, establishing a concentration gradient. The measured current decays as the concentration depletion region grows out into the solution. In chronoamperometry (10), this decaying current is followed as a function of time, and the entire transient is used to determine the rate constant at that potential. A series of experiments may be performed with varying potential step sizes so that the potential de— pendence of the rate constant can be evaluated. At large enough over- potentials, though, the rates of the electron-transfer reaction are so large that the decay is controlled entirely by the diffusion of reactant through the solution. These diffusion-limited transients contain no heterogeneous kinetics information, and the onset of dif- fusion control defines the largest measurable rate constant under those conditions. Normal pulse polarography (45) involves the same series of potential steps of increasing magnitude, but only the current flow- ing at one specified time is recorded. If these current values are 49 50 plotted against the applied potential (as is commonly done), a curve of a similar shape as a d.c. polarogram results. The data in this form can be analyzed to determine the dependence of the rate constant on potential as well. For a totally irreversible process - kf Ox + ne + Red (3.1) the equation describing the response of the ideal current-time tran- sient to a potential-step perturbation can be shown to be (44) i = nFACk f exp(kgt/D) erfc(kftk/Dk) (3.2) At the extreme of very large electron transfer rate constants, kf, the rate of decay is a simple function of time, as given by the Cot- trell equation (46), 1 - nFACDLi/ (195:5) (3 . 3) As the above equations were derived assuming an instantaneous po- tential—step perturbation, it is of interest to study the effect of finite potentiostat risetime on the results of these experiments. In addition to this instrumental nonideality, weak reactant adsorption ‘will.also be studied. 51 3.2. Analysis of Data from Potential-Step Experiments Since all the terms in Equation 3.2 are known except the rate constant and since it is possible to calculate decay transients given values of the variables, nonlinear regression suggests itself as a convenient method for extraction of the rate constant from the ex- perimental current-time transient (47). If either the concentration or the diffusion coefficient is uncertain, a two-parameter analysis may be used. It is not possible, of course, to vary all three parameters since there would be an infinite number of possible solu- tions. Other methods of analysis have been used for the extraction of rate constants from these data which involve linearized forms of Equa- tion 3.2 (48,49). These analyses are limited to particular ranges of arguments ("large", "small") of the erfc(x) term. Because non- linear regression is not subject to either the limitations or the approximations of these linear analysis methods, it was the tech- nique used for the analysis of the chronoamperometric transients in all the following work. A Laplace plane analysis has also been suggested (50) for deriva- tion of values of rate constants from chronoamperometric decay tran- sients. Its main advantages are that it is not necessary to calcu- late the exponential error function complement term, and that the transformed data are linear, so that a simple linear least squares treatment will suffice. Since it is possible to calculate this func— tion and to perform nonlinear regression quite routinely, there seems to be no advantage in transforming the data to the Laplace plane 52 for analysis. An optimal time range for the kinetics analysis of irreversible transients has not appeared in the literature. Recording data at 100 us intervals is feasible during the experiment if a high-speed transient recorder or a computer is used (51). Twenty current values recorded at this rate or at a rate 10 times slower (2 ms or 20 ms total time) were used in the following analyses. If, however, ob- viously nonideal features (iggg, a peak in the chronoamperometric decay curve) were observed at the beginning of the transient, the first several points were not used in the analysis. An alternative method of extracting rate constants from this type of data is to use the Oldham-Parry (48) method, in which the value of the current at a particular sampling time is compared to what it would have been had the transient been purely diffusion limited. This method is generally used when the data are collected as a pulse polaro- gram, where one simply takes the ratio i/i11m for currents on the rising part of the wave. Analysis of data by the latter method has several advantages. The concentration terms cancel out; uncertainty in this value does not effect the derived values of the rate constants. Additionally, measurements are easily carried out using pulse polarography, as many commercial instruments include this technique. The disadvan- tages, however, seem to outweigh these advantages in many instances. Each rate constant is generally derived from only two measurements of the current, and the derivation of any value depends on the cor- rect measurement of the limiting current, i Often, this value lim° 53 is not accessible for some reason, such as another electrode reaction obscuring the region of interest of the wave, or possibly because of instrumental nonidealities. This will be discussed later. Chronoamperometric analysis, on the other hand, bases its deriva- tion of the rate constant on many current measurements along a single transient. Adjusting the concentration or the diffusion coefficient simultaneously costs only one degree of freedom; this is easily com- pensated for by the large number of experimental points. No separate measurement of the limiting current is needed, so that access to this region of the polarogram is not necessary. The disadvantage of chronoamperometry is the somewhat more complicated instrumentation needed to record the current-time transient. The Oldham—Parry analysis can be performed on a hand-held calculator while nonlinear regression analyses require a computer to implement, but this is not a real advantage to the practicing worker as a computer system is often utilized to perform the experiment, and subsequent analysis of the data can easily be performed on-line. Tyma, ggflgl. (21) have compared the performance of the pulse polarographic analysis and the chronoamperometric analysis for fast, irreversible electrode reactions. Both procedures yielded essentially identical values of the rate constant at each potential, but it was 'noted that it was easier to spot nonideal conditions just by examin- ing the shape of the polarograms The methods had roughly the same maximum accessable rate constant. Although evidence of nonideal condi- tions was noted in this study, no attempt to analyze any apparently nonideal (peaked) polarogram was made. 54 Due to the fact that the nonlinear regression analysis of chrono- amperometric transients seems to provide the most statistically re- liable method of deriving rate constants, the following work will focus on the effect of various nonidealities when the data is analyzed in this manner. The effect of the nonidealities on the shapes of pulse polarograms and the rate constants derived from them will be examined as well. 3.3. Unique Aspects of Simulation The simulation of irreversible chronoamperometric decay transients and normal pulse polarograms is straightforward; the method outlined in Chapter 2 was used directly. The model chosen for the applied potential waveform was a simple exponential described by a time constant: E - [1 — exp(-t/T)] + Einit (3-4) Estep The actual applied potential profile of a fast potentiostat has been determined (21). The waveform was found to be described by a double exponential curve modified by a triangular deviation super- imposed on the rising part of the waveform. However, a single ex- ponential form also provides an acceptable fit and is more suitable for use here, in that only one parameter is needed to describe the risetime characteristics, rather than four for the experimentally determined profile. Since the potential is changing throughout the risetime, the 55 potential dependence of the rate constant k must be considered: f anF kf kstd exp[- RT (E - Estdn (3.5) The equation used in the simulations produced a rate constant as a function of time given a potential step size, the rate constant at the final potential, and the time constant, k =- k EXP[% E exp<-u/1-)] (3.6) f f E , step step Thus, for every new time period (every At), a new rate constant is calculated to reflect this finite potential risetime. 3.4. Effect of Finite Potential Risetime The distortions in the chronoamperometric transient produced as a result of non-ideal potential control are illustrated in Figures 3.1a and 3.1b. Figure 3.1a displays the effect when the potentio- stat time constant T is short enough so that the potential is close to the desired potential before the current is first sampled. As expected, the current is too high at every point along the curve (and, in fact, will never exactly meet the ideal line). Figure 3.lb shows the distortions produced when T is large enough that the potential is not yet near the control potential when the current is first sampled. Here we see a steep rise in current as the potential ex- ponentially approaches the final potential, and then the usual decay, with currents larger than ideal at every point. 56 200 3 100 g 1 I l “‘ 3 g _ I l ' c) 200 r 100 I l 1 0 0.5 1.0 1.5 2.0 Time (ms) Figure 3.1. Chronoamperometric transients for irreversible reac- tions illustrating the effects of potentiostat risetime, with kf - 0.3 cm/s, C - 1 mM, D - 1x10.5 cmZ/s, Estep - -500 mV. Curve 1: ideal system. Curve 2: T - 20 us. Curve 3: T - 50 us. 57 A comparison of the shapes of the ideal and nonideal transient responses suggests that a rate constant derived from the data on the basis of Equation 3.2 will appear to be larger under finite risetime conditions. In extreme cases, we might even see decay transients which appear steeper than pure diffusion control would allow. Any attempt to derive a rate constant under these conditions using the conventional diffusion model will surely meet with failure. Addi- tionally, any attempt to analyze transients with sampling times so short that a peak appears cannot be expected to yield a valid value for the rate constant. It is important to consider the effect that the time range over which the data are analyzed has on the resulting values of the rate constant. Although the effect of the nonideality upon the transient diminishes sharply with time, the sensitivity of the apparent rate constant to smaller variations in the shape of the transient in- creases with time. It is not intuitively obvious which factor, if any, will dominate, and whether the most reliable rate constant will be derived from the short or from the long time range. A set of chronoamperometric transients was simulated with various potentiostatic time constants for a number of rate constants. Table 3.1 shows the error in the rate constants derived from these tran- sients by means of one-parameter nonlinear regression analyses for a time range of 2 ms (100 us sampling time), and the same results when the current is sampled at 1 ms intervals over 20 ms. Several observations can be made. First, and most obvious, the error in the rate constant increases as the time constant increases. 58 Table 3.1. Error in Rate Constants Derived3 from Chronoamperometric Transients for Irreversible Reactionsb Due to Finite Po- tentiostat Risetime. kf (cm/s) T (us) 0.03 0.1 0.3 1.0 Short Time Rangec g 0.5 -—- +0.2Z +1.0% +7.5% E l --- 0.4 2.0 16.7 F 2 --- 0.8 4.0 43.7 5 --- 2.0 11.1 f 10 --— 4.2 27.1 f 20 +0.9Z 7.0 73.1 f 50e 3.1 18.8 f f Long Time Ranged 0.5 --- +0.12 (+0.62 (+6.02 1 --- 0.2 1.2 13.2 2 --- 0.4 2.5 33.3 5 --- 1.1 6.8 f 10 --- 2.2 14.9 f 20 +0.72 4.4 37.7 f 50 1.9 12.2 f f aOne-parameter nonlinear regression. bC - 1 mM, D a l x 10"5 cmz/s, E a -500 mV. step c100 us sampling time, 2 ms time range. dl ms sampling time, 20 ms time range. e fAnalysis failed to yield reasonable value of rate constant. First one or two points dropped before analysis. 59 Larger time constants produce more severe distortions in the shape of the transients, hence, larger errors. Second, the relative error is greater for larger rate constants at a given time constant. A come parison of the results for the two time ranges shows that there is a slightly larger amount of error for the transients sampled at short times. Although it is not a large difference, some advantage is gained by using a longer sampling time. ; Other results show that the error in the rate constant due to these instrumental nonidealities does not depend on the concentration. For values of reactant concentration ranging from 0.1 to 10 mM, the error was found to be constant, even though the current varied over 2 orders of magnitude. It was also noted that the size of the potential step has only a mild effect upon the value of the rate constant derived from the nonideal transient for a given risetime. For example, analysis of a system with T - 2 us and kf - 0.3 cm/s yielded rate constants of 0.3122 and 0.3156 for a 500 mV and a 1000 mV potential step, respectively. In an actual experiment, of course, the time constant 1 would depend on the magnitude of the potential step. It is reasonable to consider that, since the potential does not reach the desired value until some point after the beginning of the experiment, the rest of the transient would be "time-shifted" so that the apparent time zero occurs a short period after the potential begins to rise. If this were the case, the equation which describes the transient could be written 1 , nFACkf exp[t§(t-Ac)/D]er£c[kf(c-At)*/Dk] (3.7) 60 A two-parameter nonlinear regression analysis (varying kf and At) ‘might be expected to yield a more accurate value of the rate constant than could the aforementioned one-parameter analysis. Table 3.2 compares values of rate constants derived from Equations 3.2 and 3.7 for given time constants and rate constants over both the long and short analysis time ranges. These results indicate that an improvement in the accuracy of kf is indeed obtained by the two-parameter analysis in most cases; the correction is, however, not perfect, and varies with experimental parameters. For larger rate constants (0.3 and 1.0 cm/s) in the 20 ms transients, hardly any effect is seen. The greatest improvement is found at smaller rate constants and shorter analysis time ranges. The correction is not perfect because the effect of finite rise- time is not simply to "time-shift" the transient to later times; the concentration gradient begins to form, and the surface concentration begins to change, as soon as the potential rises enough to allow the electrode reaction to proceed at a significant rate. Thus, Equation 3.7 is inadequate because it implies an infinitely fast potential step at some time At after time zero, and an unperturbed system prior to the step, which is clearly not the case here. An examination of Figure 3.1a or 3.1b shows that the difference in time values at a given current value varies with time throughout the transient. It does not seem likely that the "time-shift" analysis will be of much utility to the electrode kineticist. The greatest gains in accuracy are obtained under conditions such that the error in the apparent rate constant is small anyway. Additionally, the effect of 61 Table 3.2. Comparison of Results of Standard and "Time-shift" Analysesa of Chronoamperometric Transientsb with Finite Risetime. Error in kf Short Time Rangec Long Time Ranged T (us) kf(cm/s) Standard Time-Shift Standard Time-Shift 2 0.1 +0.78% +0.02% '+0.422 +0.04% 0.3 4.1 0.4 2.5 2.3 1.0 43.7 42.1 33.3 31.4 5 0.1 2.0 0.06 1.0 0.08 0.3 11.1 1.4 6.7 6.3 10 0.1 4.2 0.18 2.2 0.15 0.3 27.1 4.2 14.8 15.2 20 0.03 0.92 0.06 0.73 0.01 0.1 7.0 5.9 4.4 0.30 0.3 73.1 29.4 37.8 38.3 aStandard analysis: one-parameter (kf) nonlinear regression. "Time- shift" analysis: two-parameter (kf and At) nonlinear regression. b 5 cm2/s, Estep I -500 mV. c100 us sampling time, 2 ms time range. d c-1mM, D=1x10' 1 ms sampling time, 20 ms time range. 62 such an empirical correction would be difficult to predict under actual experimental conditions, where finite potential risetime is not the only nonideality encountered. 3.5. Effect of Weak Reactant Adsorption The effect of weak adsorption on the shape of the chronoampero- metric transient depends on the relative degree of adsorption before and after the potential-step perturbation. Figure 3.2a shows the deviation from ideality for k - 0.1 cm/s when there is an equal amount f of adsorption before and after the step. Currents are greater for all times; this is a consequence of the surface concentration enhance- ment, and the resulting faster reaction rate. Figure 3.2b displays the shape of the transient which results under differing degrees of adsorption before and after the step, again with kf - 0.1 cm/s. When there is adsorption at the initial potential, but none at the final potential, currents are now even higher than the previous case at short times because the adsorbed species which are released at the onset of the potential step enhance the concentration at the surface and reduce reactant depletion in the diffusion layer. The effect of this "extra" reactant is seen throughout the transient because of the cumulative response of the concentration gradient. The opposite ef- fect is seen when the reactant only adsorbs on the electrode after the step. The current is seen to be too small at short times because the additional flux from the diffusion layer needed to "coat" the surface depletes reactant in the vicinity of the electrode. Once the ad- sorbed layer is completed (which is a diffusion-controlled process 63 Current (uA) 0 0.5 1.0 1.5 2.0 Time (ms) Figure 3.2. Chronoamperometric transients illustrating the effects of weak reactant adsorption, with parameters as in Figure 3.1, except kf - 0.1 cm/s. Curve 1: ideal system. Curve 2: Kid a 5 a _ 0 1 a -5 B ' Kad 2x10 cm. Curve 3. Kad 2x10 cm, Kad 0. Curve 4. 5 1 _ Kad = 0, Kad = 2x10 cm. 64 because of the rapidly attained equilibrium), the reaction proceeds with the enhanced surface concentration, which will tend to increase the total current. This effect becomes pronounced at longer times, when the current is seen to exceed the ideal value. In summary, the chronoamperometric decay is steepest when the reactant is adsorbed before the potential step (but not after), and mildest when it is adsorbed only after the perturbation. One would expect larger apparent rate constants from steeper decay curves; this is indeed observed upon analysis of these nonideal transients. Some chronoamperometric transients were simulated with various values of kf and adsorption coefficients. The results of the non- linear regression analyses of these transients according to the ideal decay equation (Equation 3.2) are given in Table 3.3. Somewhat greater errors in the rate constant are observed for the short-time analyses, but the difference is not large. The major difference between the two cases, in fact, is that the short time results tend to conform to the earlier, intuitive analysis (112;, the concept of the rate of decay being directly proportional to the rate constant) than do the long- time results. At longer times (and higher rate constants) the derived values of kf are found to be the largest when the degree of adsorp- tion is equal before and after the step. This variation in the results as a function of the time period over which the transient is analyzed is due to the fact that the ideal chronoamperometric decay equation does not exactly fit the shape of the decay when adsorption is present for any value of kf. It was seen in Figure 3.2b that after a sufficient period of time, the 65 Table 3.3. Error in Rate Constants Derived3 from Chronoamperometric Transientsb Due to Weak Reactant Adsorption. Adsorption Condition Before Step After Step Before + After kf (cm/s) only‘2 only 3 tepe Short Time Rangef 0.01 +15% -12.8% +2.02 0.03 19 -12.3 5.6 0.10 34 -1l.6 22 0.30 86 - 2.6 190 Long Time Range8 0.01 +6.01 - 3.9% +2.02 0.03 9.0 - 3.0 6.3 0.10 18 + 3.0 26 a One-parameter nonlinear regression. 5 5 cmzls, E = -500 mv. c-1mM, 0a1x10' step cK1 - 2 x 10"5 cm, K - 0 cm. ad ad 1 -5 dKad - 0 cm, Kad - 2 x 10 cm. i -5 8K,ad - Kad - 2 x 10 cm. f100 us sampling time, 2 ms time range. 81 ms sampling time, 20 ms time range. 66 current in the nonideal experiment becomes higher than the ideal value. If the analysis looked only at this region of the curve, one might indeed expect a derived rate constant to be too high under these conditions. That the derived rate constants are found to be highest at equal reactant adsorption before and after the step can be understood upon observation of Figures 3.2a and 3.2b. Although the current begins much higher in Figure 3.2b, it returns much faster to the ideal value than does the transient in Figure 3.2a, where equal adsorption before and after is present. At longer times, the devia- tion from ideal is smaller when there is adsorption only before the step, which would account for the results in Table 3.3. A more detailed view of the error in the derived rate constant as a function of the adsorption coefficients is given in Table 3.4. The complete variation for a typical rate constant of 0.1 cm/s analyzed over a 2 ms time range is shown. It is obvious that more error is produced by more reactant adsorption, although the error becomes for- tuitously small for the conditions where the derived rate constant goes from too high to too low. This point at which the deviations exactly compensate to yield the correct rate constant seems to depend on too many factors to isolate. Another question which could be asked is whether the error in the rate constant depends only on the amount of adsorbed reactant, F, or on the ratio of adsorbed species to bulk species, Kad' A series of transients were generated in which the bulk concentration and ad— sorption coefficients were varied in such a way as to keep the surface excess constant. The resulting rate constants showed wide variation 67 Table 3.4. Error in Rate Constants Derived3 from Chronoamperometric Transientsb with Varying Amounts of Weak Reactant Adsorp- tion. Kad (cm) Kid (cm) 0 1 x 10"6 3 x 10'6 1 x 10'5 2 x 10‘5 0 0 -0.42 -1.3z -5.22 -11.62 1 x 10"6 +1.42 +1.0 +0.2 -3.7 -1o.2 3 x 10‘6 4.3 3.9 3.1 -0.7 - 7.4 1 x 10’5 15.2 15.0 14.3 +10.7 + 3.4 2 x 10'5 33.6 33.7 33.4 30.2 22.0 aOne-parameter nonlinear regression. b100 us sampling time, 2 ms time range. k a 0.1 cm/s, C = 1 mM, -5 2 f D I l x 10 cm /s, E I -500 mV. step 68 in the amount of the error, with low concentrations and high adsorp- tion coefficients producing the largest error. Another series of transients were simulated again with varying concentration, but with constant adsorption coefficient (care was taken not to exceed a sur- face excess greater than about 102 surface coverage). The rate constants derived from these transients were identical, showing that it is the adsorption coefficient, and not the surface excess itself, which dictates the amount of error in the derived rate constants. This is expected given the linear nature of the adsorption isotherm. 3.6. Effect of Coupled Risetime/Adsorption Weak reactant adsorption and finite potential risetime combine to produce transients which yield rate constants that are in error by an amount which is somewhat greater than would be expected assuming the individual contributions were additive. There seems to be no general rule, however, to allow the prediction of the extent of this coupling. Table 3.5 shows the effects of an increasing risetime on each of the three adsorption cases, in addition to that when no adsorption is present. As expected, increasing risetime increases the error in the rate constant, but to a somewhat greater extent than when no ad- sorption is present. The same trends are seen for analyses in both time ranges. It is interesting to note the compensation of errors produced by finite risetime (which yields positive errors) and the case in which the reactant is adsorbed after the application of the potential step (negative errors). The adsorption is seen to be the controlling 69 Table 3.5. Error in Rate Constants Derived3 from Chronoamperometric Transientsb Due to Both Finite Risetime and Weak Reactant Adsorption. Adsorption Conditions T (us) None Before Onlyc After Onlyd Before and Aftere Short Time Rangef 0 0 +342 -1l.6Z +22% 5 +22 43 -11.0 24 10 4 51 —10.3 28 20 7 64 - 9.8 32 Long Time Range8 0 0 +18% +32 +26% 5 +12 22 3 28 10 2 25 4 30 20 4 30 5 34 50 12 48 12 50 a One-parameter nonlinear regression. kf - 0.1 cm/s, C 1 mM, D l x 10 cm ls, Estep . —500 mV. °1<1 - 2 x 10'5 cm, K - 1 x 10'11 cm. ad ad 1 -11 -5 dKad 1 x 10 cm, Kad - 2 x 10 cm. 1 -5 eKadIKadIleO cm. f 100 us sampling time, 2 ms time range. 81ms sampling time, 20 ms time range. 70 factor, with the error in the derived rate constant changing only slightly with increasing risetime. It is possible that the depletion of the solution near the electrode makes the experiment insensitive to the exact risetime conditions. The opposite adsorption scheme, in which the reactant is released into the solution at the time of the step, shows the largest sensitivity to the potential risetime, probably due to the opposite factors which are Operating above. When the adsorption coefficients remain constant throughout the experiment, the errors induced by each nonideality are almost additive. 3.7. Effect of Nonidealities on Normal Pulse Polarography Finite potential risetime and weak reactant adsorption can have a clearly visible effect on the shapes of pulse polarographic waves when the current is sampled at short enough times. Two things need to be considered when data is collected or displayed in this format, the first and more important of which is the apparent value of the limiting current. The kinetics analysis of the polarogram depends on this value, as do electroanalytical determinations. The second thing to be considered is the actual shape of the wave; deviations could yield nonlinear plots of in kf Kg, potential (Tafel plots) or incorrect values of alpha, the transfer coefficient, as well as simple errors in the rate constant. Figure 3.3a is a comparison of an ideal normal pulse polarogram sampled at 100 us and the corresponding polarogram which includes the influence of a finite potential time constant of 20 us. The most 71 S H H \ I4 I l 3 - 0.0 14 l l I .1 -500 -700 -900 Potential (mV) Figure 3.3. Pulse polarograms illustrating the effects of 5 cmzls. finite risetime, with a - 0.5, c - 1 mM, D - 1x10“ Curve 1: ideal system, 100 us sampling time. Curve 2: T I 20 us, 100 us sampling time. Curve 3: ideal system, 1 ms sampling time. Curve 4: r I 20 us, 1 ms sampling time. 72 noticeable feature in the nonideal polarogram is the presence of a peak, similar in appearance to d.c. polarographic maxima. This peak is disturbing because the current does not decay to the true dif— fusion-limited value (or even to a constant value) even at potentials up to 500 mV or more past the peak. Additionally, the current sampled along the wave is too high beyond about one third of the way up the wave. These deviations are very sensitive to the sampling time at which the polarogram is recorded. Figure 3.3b shows the identical system and risetime of the previous figure, except that the sampling time is 1 ms. Although the peak can still be discerned, the maximum error is only about 22, compared to about 35% when sampling at 100 us. There is much less error in the rising part of the wave as well. A problem that arises when one attempts to derive rate constants from these nonideal polarograms is that it is unclear what value of the limiting current to use, since no constant diffusion-limited value can be obtained reasonably near the wave. A comparison between the two data analysis methods for the finite risetime case would be of interest, though, so it was decided to use the current several hundred millivolts from the wave as the limiting current in the analysis. Table 3.6 shows the results of the pulse polarographic analysis with nonideal currents sampled at 100 us and the results of the correspond- ing chronoamperometric analyses. One can see that at the lower rate constants there is more error induced by the polarographic analysis, while the reverse is true for larger rate constants. One might 73 Table 3.6. Comparison of Results from Pulse Polarographic and Chrono- amperometric Analyses of Transientsa with Finite Risetimeb. Error in kf Pulse c Chrono- Estep (mV) kf (cm/s) Polarographic amperometric -450 0.0066 -9.12 O -500 0.017 -5.9 0 -550 0.045 -4.4 +2.22 -600 0.12 0 -8.3 -650 0.30 +3.3 30 3C I 1 mM, D = l x 10-5 cm2/s bT I 20 us. cSampling time I 100 us, Oldham—Parry analysis. leO us sampling time, time range 2 ms. One-parameter nonlinear re- gression. 74 expect larger errors from the OldhamrParry analysis (48) simply be- cause the deviations in the transient are highest at the shortest times; the range of sampling times for the chronoamperometric analysis extended to 2 ms, while the polarographic analysis was limited to the values at 100 us. Additionally, the use of a limiting current value which is too high will cause errors in relatively undisturbed regions of the wave. At higher rate constants, however, there seems to be a compensation effect. The measured current and the limiting current are in error by roughly the same amount, so that the i/i11m value is .closer to ideal, and less error in the rate constant results. Weak reactant adsorption also produces peaks in normal pulse polarograms. Figure 3.4a displays an ideal polarogram sampled at 100 us and one in which the reactant is adsorbed equally before and after the potential step. Note that the maximum has a different shape than that due to potential risetime; its maximum value comes at lower overpotentials, and, more importantly, the current falls to a true diffusion-limited value one or two hundred millivolts beyond the peak. Electroanalytical applications of normal pulse polarography will not suffer from this nonideality (unless the limiting region of the wave is not accessible for some reason), but errors will arise for elec- trode kineticists since the current is considerably different from ideal along all but the foot of the wave. As with the finite rise- time case, the effect of the nonideality falls off rapidly with in- creasing sampling time. Figure 3.4b illustrates this point, showing the same system as in Figure 3.4a except that the sampling time was 1 ms. 75 i / 111m -500 -700 -900 Potential (mV) Figure 3.4. Pulse polarograms illustrating the effects of reac- tant adsorption, with a I 0.5, C I 1 mM, D I 1x10.5 cm2/s. Curve 1: ideal system, 100 us sampling time. Curve 2: Kid I Kad I 2x10.S cm, 100 us sampling time. Curve 3: ideal system, 1 ms sampling time. Curve 4: Kid I Kad I 2x10-5 cm, 1 ms sampling time. 76 It is interesting to note the effects of varying amounts of ad- sorption at the initial and final potential. Figure 3.5 shows an ideal polarogram, one in which there is adsorption only at the final poten- tial, and one where adsorption is present only before the step. The latter curve starts out with currents which are too low, the former has larger than ideal currents at the foot of the wave. These curves converge towards the top of the wave to yield identical, somewhat high responses, although with smaller peaks than the equal adsorption case. Since the true limiting currents are accessible for these weak adsorption polarograms, a kinetics analysis may be carried out to determine the amount of error in the rate constants derived from the nonideal waves. Table 3.7 lists the rate constants derived from these polarograms at selected potentials along the wave, as well as the cor- responding values from the chronoamperometric analysis. At lower rate constants, the error is clearly larger for the pulse polarographic analysis when the adsorption is not equal before and after the step. At equal adsorption conditions, very little difference exists between the methods. At higher rate constants, this difference becomes negligible. It is easy to understand why this occurs. The pulse polarographic analysis uses only the current sampled at 100 us to estimate the rate constant, while the chronoamperometric uses that point plus many others at longer times. It has already been shown that the deviations from ideal are largest at short times, so an analysis based only on the shortest time cannot be expected to yield as reliable a value as would one based on many data at longer times. 77 I I F T l 1.0 - p 5' H w-t \ 0.5 - H 1 1 1 1 0 1 -500 -700 -900 Potential (mV) Figure 3.5. Pulse polarograms illustrating the effects of reac- tant adsorption, with a I 0.5, C I 1 mM, D I 1x10.5 cm2/s, 100 us sampling time. Curve 1: ideal system. Curve 2: Kid I 2x10.5 cm, .1. _ -5 Kad I 0. Curve 3. Kad 0, Kad 2x10 cm. 78 Table 3.7. Comparison of Results from Pulse Polarographic and Chrono- amperometric Analyses of Transients with Weak Reactant Adsorption. Error in kf Adsorption Conditions Before Onlyb After Onlyc Before and Afterd e f e f e f E (mV) kf (cm/s) PP CA PP CA PP CA -450 0.0066 +32% +152 -332 -122 0 +2Z -500 0.017 35 18 -29 —12 0 6 -550 0.045 38 22 -31 ~13 +6.62 8.9 -600 0.12 42 38 -28 -12 17 25 -650 0.30 83 87 -13 - 3 160 190 aC I 1 mM, D I l x 10"5 cm2/s. bKi = 2 x 10.5 cm, K a l x 10_11 cm. ad ad cxi =- 1 x 10"11 cm, K - 2 x 10'5 cm. ad ad d i f -5 Kad Red 2 x 10 . e01dham-Parry analysis, 100 pa sampling time. f1 parameter nonlinear regression, 100 us sampling time 2 ms time range. 79 An analysis of polarograms sampled at 1 ms, although the largest accessible rate constant was not as high, was found to produce sig- nificantly less error than the 100 us waves. The error observed was still fairly high, averaging around 1002, and followed the same trends as the shorter time analysis. The errors were roughly comparable to the chronoamperometric analyses; the current at 1 ms was about in the middle of the current-time transients which were analyzed. Thus, it appears that no increases in accuracy can be gained through the use of the pulse polarographic over the chronoamperometric data analysis when weak adsorption is present. Both procedures seem to suffer from roughly the same relative amount of error in the rate constant under these conditions. 3.8. Correlation of Model with Experimental Data The types of deviations noted in the previous sections have been observed experimentally in this laboratory (21). It was of interest, therefore, to try to correlate this model of finite potentiostat risetime and weak reactant adsorption with these data in an attempt to derive meaningful values of the rate constant from apparently non- ideal transients when ideal analyses are unable to do so. A set of chronoamperometric transients was available for Cr(OH2)g+ reduction on a mercury electrode (21). When displayed in pulse polaro- graphic format, a maximum was observed in the wave at short sampling times (less than about 500 us). These data were analyzed with the ideal chronoamperometric (nonlinear regression) analysis and by the pulse polarographic (OldhamrParry) method. As can be seen in 80 Table 3.8, at rate constants above about 0.16 cm/s, points deviated considerably from the Tafel line extrapolated from smaller rate constants, suggesting a failure of the analysis to derive a meaningful value of the kinetics parameter. Below this level, however, there was good agreement between rate constants produced by the two analysis methods. An alternative analysis method was then used to attempt to extend the range of accessible rate constants under these nonideal conditions. Instead of the ideal equation, a parallel simulation routine was used as the calculation subroutine in the nonlinear regression program. However, there are now four unknowns in the equation: the rate constant, the risetime of the potentiostat, and the adsorption co- efficients before and after the application of the step. Since the potential was stepped from a point at which the adsorption coefficient is predicted to be quite small (see Chapter 1), it was assumed that any diffuse layer adsorption at this potential would be negligible, (allowing one of the unknowns to be eliminated. The results of the analysis show very good agreement with the two ideal analyses at the lower rate constants; at higher values, however, the derived values are considerably closer to the extrapolated line than either of the other two methods, as shown in Figure 3.6. Indeed, correct rate constants up to 1.47 cm/s seem to have been extracted from these data. There was, however, some problem with two of the intermediate transients; the reason for this is unknown. Table 3.9 contains the resulting values of all three of the parameters. Although the rate constant is derived successfully, there seems to be little sense to 81 Table 3.8. Rate Constants for Cr%:§) Reduction Derived by Pulse Pol— arographic and Chronoamperometric Analyses. kf (cm/s) Pulse Chrono- E (mV) Polarographic amperometric -975 0.0109 0.0112 -1000 0.0180 0.0189 -1025 0.0355 0.0329 -1050 0.0535 0.0560 -1075 0.0945 0.0968 -1100 0.15 0.185 -1125 a 1.464 -1150 a 0.589 -1l75 a b -1200 a b aCurrent was too close to diffusion-limited value to obtain meaningful results. bAnalysis failed to yield meaningful results. 82 0 A Q \ B U V '44 .3 9. ‘1 an O H -2 ‘ 1 -1.0 -1.1 -1.2 Potential (V vs. s.c.e.) Figure 3.6. Rate constants for Cr(OH2)2+ reduction derived from experimental chronoamperometric transients using simulation analysis, vs. potential. Line is least-squares fit of lower six points. 83 Table 3.9. Results of Simulation Analysis of Cr?:q) Data. Derived by Analysis E (mV) kf (cm/S) T (118) Kad (cm) -975 0.0112 11.5 0 -1000 0.0190 31.5 2.5 x 10‘6 -1025 0.0331 8.6 1.8 x 10'6 -1050 0.0554 33.9 7.0 x 10'6 -1075 0.0955 20.7 11 x 10'6 -1100 0.168 20.7 180 x 10"6 -1125 0.494 11.2 0 -1150 1.321 9.4 139 x 10‘6 -1175 0.904 47.5 0 -1200 1.47 52.0 12 x 10'6 84 be made of the series of adsorption coefficients and risetimes. It is possible that either nonideality can account for the deviations in the transients; one factor may dominate on the basis of trivial differences in the curves. It is also possible that the exact shape of the transient is extremely sensitive to the shape of the profile of the applied poten- tial. A single exponential model might not be sophisticated enough to adequately fit these data. If this is the case, the actual applied potential profile can be recorded together with the current response to eliminate errors due to the incorrect choice of a risetime model. Clearly, much more remains to be done in this work. The results of these preliminary experiments are promising, but further systems need to be studied under more well-defined conditions. The major drawback to the use of this method is the long computation time re- quired for this type of analysis - approximately three orders of magnitude longer than the ideal nonlinear regression analysis. This is clearly not the method of choice for data on which the ideal analyses perform well. CHAPTER 4 POTENTIAL-STEP EXPERIMENTS FOR THE STUDY OF QUASI-REVERSIBLE ELECTRODE REACTIONS 85 4.1. Description of Experiment The potential-step experiment for quasi-reversible reactions (34) is analogous to that described in Chapter 3 except that the chemical system to which it is applied now includes a back reaction component: _ kf Ox + ne 2 Red (4.1) “6 Both large and small potential steps may be used to study the kinetics of this type of process. In small-step experiments (10), both the oxidized and reduced species are present in the solution while the electrode is maintained at the equilibrium.potential. The applied potential is suddenly changed by several millivolts, and the current which flows at this new potential is monitored. Large-step experiments (10) start with only one species of the redox couple in a solution in which the electrode is held at a poten- tial such that this species is strongly favored on the basis of the Nernst equation. A potential step of several hundred millivolts is applied, causing the reaction to proceed. The current is again moni- tored as a function of time. The results of these large-step experi- ments can be treated in the same way as the small-step experiments, or alternatively displayed and analyzed in the form of a normal pulse polarogram. The equation which describes the current response of a quasi- 86 87 reversible system to either a large or small potential-step perturba- tion is (34): — 2 2 . 1 - nFA (kaOX-kbcred)exp(kft/Dox+kbt/D ) red erfc(kftk/DEX-t-kbtk/Difed) (4.2) Note that when kstd is very small, the reverse reaction terms drop out, yielding the equation for the irreversible case. This equation describes the ideal experiment in which there is no reactant adsorbed at the electrode surface, and in which the potential rises instantaneously to the new value. It is of interest, therefore, to determine the effects of reactant adsorption and finite potentiostat risetime on rate constants derived from these potential- step experiments, both in chronoamperometric small- and large-step procedures, and to compare the large-step chronoamperometric data treatment to that of normal pulse polarograms. 4.2. Conventional Data Analysis As was the case for irreversible redox systems, nonlinear regres- sion on Equation 4.2 is an obvious method with which to derive hetero- geneous rate data. The rate information of interest is contained in the standard rate constant, from which kf and kb can be derived. Thus, it is possible to perform a one-parameter nonlinear regression analysis of the current-time transient to derive a value of the standard rate constant. Since the experiment is (ideally) carried 88 out at constant potential, the forward or reverse rate constant could be obtained instead. (This is not the case for coulostatics and gal- vanostatic double pulse experiments in which only the standard rate constant is accessible.) Normal pulse polarography (52) can also be performed on quasi- reversible systems. Now, however, the OldhameParry analysis (48) must be modified to include the effect of the back reaction (46). Since only forward rate constants are derived through this procedure, the Estd of the redox couple must be known in order to determine the standard rate constant (this is also the case for the small-step nonlinear regression analysis). The advantages and disadvantages of each type of analysis as described in Chapter 3 apply for quasi-reversible reactions as well. 4.3. Unique Aspects of Simulation The simulation of quasi-reversible systems was a straightforward extension of the irreversible simulations described in Chapter 3. A single exponential was used as the applied potential waveform, and the Henry isotherm was applied independently to both the oxidized and reduced form of the species. In small-step experiments, it is not necessary to consider a po- tential dependence of the adsorption coefficients, so that values of Kox and Kred at the equilibrium potential are sufficient to describe the adsorption. The large-step experiments, however, need values of the adsorption coefficient for each Species both before and after the step as these parameters exhibit a dependence on potential. 89 The parallel simulation method was used for simulations of small- step experiments. Conventional simulations were used for pulse pol- arographic and large-step experiments because the original goal of these calculations was simply to observe the morphology of the wave. In determining the error in rate constants derived from these transients and polarograms, the values of the rate constants obtained by the same analysis procedure on ideal and nonideal transients were compared to help eliminate the effects of simulation error. 4.4. Effect of Risetime on Small-Step_Chronoamperometry Finite potentiostat risetime produces very similar effects in the shape of small-step chronoamperometric transients from both ir- reversible and quasi-reversible systems. Figure 4.1 shows an ideal transient together with one generated with a potential risetime con- stant of 50 us. Because of the relatively long risetime, the poten- tial has not yet reached the desired value by the time the current is first sampled; the shape of the nonideal transient at short times illustrates this. Once the potential is at the desired value, the nonideal transient is larger than ideal, and will remain larger be- cause the slow potential rise causes the diffusion layer around the electrode to be less depleted than it would have been had the experi- ment been ideal. A series of small potential-step experiments was simulated using various standard rate constants and potential risetime constants. Two time ranges were used, as for the irreversible systems. The errors in the rate constants derived from these simulated transients using a one-parameter nonlinear regression analysis are listed in 90 40 h) C Current (uA) 20 0 0.5 1.0 1.5 2.0 Time (ms) Figure 4.1. Chronoamperometric transients for quasi-reversible reactions illustrating the effects of finite risetime, with kstd I I 1 mM, D I D I OK 0.1 cm/s, a I 0.5, E I -10 mV. Cox ' C red step red 5 1x10- cm2/s. Curve 1: ideal system. Curve 2: T I 50 us. 91 Table 4.1. These results show that serious errors in the rate constant are found for systems with kstd greater than about 0.1 cm/s with reasonable time constants of 5 or 10 us, regardless of the time range used. A comparison of the results for the two time ranges shows that the short time range analysis yields more accurate results under these conditions. This is the opposite of what is found for the analysis of irreversible reactions, in which larger time ranges pro- duced less error, although the degree of this error is roughly com- parable overall. ‘An adjustable experimental parameter in this method is the size of the potential step which is applied to the system, and it is of interest to determine if it is possible to adjust this value to op- timize the measurement of the rate constant under nonideal conditions. A series of transients was generated with a range of step sizes from 0.5 mV to 50 mV, keeping all other parameters constant. The error in the rate constants derived from these transients was independent of the step size, except that the error increased dramatically for step sizes over about 30 mV. (In actual practice, however, the time constant probably depends on the size of the potential step.) It is also of interest to determine whether the concentrations of the reactants affect the accuracy of the derived rate constant under constant risetime conditions. Several series of transients were generated for various values of Cox and Cred in a range from 0.1 mM to 10 mM. When the concentration of each of the species was equal, the error in the derived rate constant was independent of the 92 Table 4.1. Error in Rate Constants Deriveda from Chronoamperometric Transients for Quasi-Reversible Reactionsb Due to Finite Risetime. kstd (cm/8) T (us) 0.03 0.1 0.3 1.0 Short Time Rangec 1 +0.12 +0.42 +2.32 +292 2 0.1 0.8 4.7 129 5 0.3 2.0 13.4 e 10 0.7 4.3 35.8 e 20 1.3 9.0 215 e d Long Time Range 2 +0.12 +0.52 +3.82 +1122 5 0.2 1.2 10.4 e 10 0.4 2.4 24.7 e 20 0.7 5.1 88.8 e 50 1.9 14.7 e e aOne-parameter nonlinear regression. b -5 2 a _ Cox I Cred I 1 mM, Dox I Dred I l x 10 cm /s, Estep 10 mV. c100 us sampling time, 2 ms time range. d1 ms sampling time, 20 ms time range. eAnalysis failed to yield reasonable value of rate constant. 93 concentration. There was, however, a dependence on the ratio Cox/ C More error was induced in the rate constant when cox/Cred red' deviated from unity in either direction. Even though the actual size of the deviation in the transient wasn't strongly affected by this variation, there simply is less kinetics information available in the transients as the equilibrium potential gets farther from the standard potential. Thus, the kinetics parameter is more sensitive to deviations in the shape of the transient, and larger errors will result for a given size deviation under these conditions. 4.5. Effect of Adsorption on Small-Step Chronoamperometry The influence of weak reactant adsorption on the shape of the small-step chronoamperometric decay curve is shown in Figure 4.2. The transient which was generated with both species of the redox couple adsorbed is higher than ideal all along the transient, espec- ially at small times. When only one of the species is adsorbed, the deviation from ideal is roughly half that seen when both species ad- sorb; there is only a minor difference in the shapes of the transients with only the reactant or product adsorbed. This additional current is due to the reaction of the adsorbed species and to the steeper concentration gradient which is needed to replenish the adsorbed layer. To obtain an overview of the accuracy of the rate parameters derived fll'om transients generated with weak reactant adsorption, a series of <fllrves were generated with equal adsorption of 0x and Red for various rate constants and adsorption coefficients. The results of the 8.Ilalysis of these transients are shown in Table 4.2. There is 94 50 I ' ' 40 - 2 4/ 3 U f": ,, 30 - :3 U 1 20 - l l L 0 0.5 1.0 1.5 Time (ms) Figure 4.2. Chronoamperometric transients illustrating the effects of reactant adsorption, with kstd I 0.1 cm/s, a I 0.5, E I -10 mV, C I C 5 cmF/s. ox step I 1x10 red . 1 mM, Dox - Dred Curve 1: ideal system. Curve 2: K I K 5 ox re d I 2x10 cm. 95 Table 4.2. Error in Rate Constants Derived3 from Chronoamperometric Transientsb Due to Weak Reactant Adsorption. kstd (cm/s) xox - Kred (cm) 0.03 0.1 0.3 1.0 Short Time Rangec 1 x 10"6 +0.62 +2.12 +6.42 +28.32 3 x 10‘6 1.9 6.5 23.4 e 1 x 10’5 6.2 25.2 e e 2 x 10'5 12.4 62.7 e e a Long Time Range 1 x 10'6 +0.62 +2.02 +6.52 +28.52 3 x 10'6 1.9 6.3 24.1 e 1 x 10‘5 6.5 26.7 e e 2 x 10'5 13.8 93.6 e e aOne-parameter nonlinear regression. Cox Cred 1 mM, Dox Dred 1 x 10 cm /s, Estep 10 mV. c100 us sampling time; 2 ms time range. dlms sampling time; 20 ms time range. eAnalysis failed to yield reasonable value of rate constant. 96 virtually no difference between analysis of 2 ms or 20 ms transients, except at the largest levels of adsorption. The error in the de- rived rate constants is quite severe for small amounts of adsorption (Kox I K.re I 3 x 10-6 cm, or a coverage of about 12 of a monolayer d in a 1 mM solution) when the rate constant is greater than about 0.1 cm/s. The accuracy of the rate constant determination depended only to a very small extent on the size of the potential step which is applied to the cell, as was the case with finite risetime. A potential step size of 10 mV was used for all small potential-step simulations. It is of interest to expand the study of weak reactant adsorption to situations in which Kon is not equal to Kred' Table 4.3 shows the results of the analysis of a series of simulated transients for various values of the adsorption coefficients at a rate constant of 0.1 cm/s. It can be seen that the error increases roughly with the total amount of adsorption, although the effect is not additive for both species. Combined adsorption of both species produces more than the sum of the errors caused by adsorption of the individual species. The slight asymmetry about the Kox I Kred diagonal reverses with a potential step of the Opposite sign; apparently the adsorption of the reactant species produces larger deviations in the shape of the transient than does adsorption of the product. Variation of the relative concentrations of the oxidized and re- duced half of the redox couple while keeping the adsorption coefficients constant has the same sort of effect as in the finite risetime case. Table 4.4 shows the results of the analysis of a typical series of 97 Table 4.3. Error in Rate Constants Derived3 from Chronoamperometric Transientsb Due to Weak Reactant Adsorption (K i K ). ox red K'red (cm) Kox (em) 0 1 x 10'6 3 x 10'6 1 x 10‘5 2 x 10'5 0 -- +0.82 +2.52 +8.82 +18.12 1 x 10‘6 +1.22 2.1 3.8 10.3 19.8 3 x 10‘6 3.8 4.7 6.5 13.3 23.4 1 x 10‘5 13.6 14.6 16.9 25.2 37.9 2 x 10"5 29.3 30.7 33.7 45.0 62.7 a One-parameter nonlinear regression. b C. kstd I 0.1 cm/s, C I C 100 us sampling time; 2 ms time range. I 1 mM, D I l x 10- cmz/s, ox red ox Dred E I -10 mV. step 98 Table 4.4. Error in Rate Constant Derived3 from Chronoamperometric Transientsb Due to Weak Reactant Adsorptionc (C i C ). ox red Cred (mM) (mM) 0.1 0.3 1 3 10 ox 0.1 +2.12 +2.62 +4.22 +7.22 +14.22 0.3 2.2 2.1 2.7 4.2 7.6 l 3.1 2.2 2.1 2.6 4.2 3 4.9 3.0 2.2 2.1 2.7 10 9.0 5.1 3.0 2.2 2.1 aOne-parameter nonlinear regression. b 100 us-gampéing time, 2 ms time range. kStd I 0.1 cm/s, Dox I Dred 1 x 10 cm /s, E I -10 mV. . step c at I1x10-6cm. ox red 99 nonideal transients with various concentrations Cox and Cred' Again, the Cox/cred ratio controls the accuracy of the derived rate constant, with a slight asymmetry around the Cox I C diagonal. red These results rule out the possibility of minimizing errors due to weak reactant adsorption by adjusting the concentration of the redox couple. If diffuse-layer adsorption is occurring, however, it may be possible to shift the potential at which the experiment is performed closer to the electrode's p.z.c., sacrificing some sensi- tivity to the electrode kinetics to lower the extent of the adsorp- tion. The success of this strategy depends on the specific chemical system involved, however. 4.6. Comparison of the Effect of Risetime on Normal Pulse Polarography and Large-Step Chronoamperometry A finite potentiostat risetime affects the normal pulse polarogram of a quasi-reversible system in much the same manner as is seen for irreversible reactions (Chapter 3). Figure 4.3 shows some simulated pulse polarograms of a species with kstd I 0.1 cm/s at a 100 us sampl- 'ing time with the time constant of 20 us, 10 us, and 0 us (ideal case). Again, the current along the nonideal waves is too large at all po- tentials, with the least deviation at the foot of the wave and far out into the diffusionrlimited region. The shape of the deviation is virtually independent of the value of the standard rate constant" of the system as can be seen by comparing Figures 3.3a and 4.3. Other aspects of the shape of the nonideal waves show the same trends as were seen in the previous chapter. The true value of the 100 400 b 3\ .. '\ A 2 1 3 u 8 H s 200 - U 100 - / 0 I l 1 1 1 0 -200 -400 Potential (mV) Figure 4.3. Pulse polarograms illustrating the effects of finite risetime, with kstd I 0.1 cm/s, a I 0.5, Estd I 0 mV, Cox I I5 2 1 mM, Dox I Dred I 1x10 on Is, 100 us sampling time. Curve 1: ideal system. Curve 2: T I 10 us. Curve 3: T I 20 us. 101 limiting current is not approached until many hundreds of millivolts past the wave. The observed current is decreasing continuously past the peak.although this might not be apparent in the presence of an increasing baseline current. There is also a strong dependence on sampling time, with deviations becoming considerably smaller than those in Figure 4.3 when the current is sampled at 1 ms. The kinetics analysis of these polarograms is complicated by the absence of any apparently normal limiting current region. However, it is possible to use the current at an arbitrary potential after the peak so that results of the pulse polarographic and chronoamperometric analyses may be compared. (Since the choice of the limiting current is arbitrary, though, any analysis based on it must be equally ar- 1 bitrary.) Several transients were simulated at various points along the wave for a standard rate constant of 0.1 cm/s and a time constant of 20 us. Ten points along each transient were recorded, equally spaced from 100 us to 1 ms. Two polarograms were assembled from these data, representing the extremes of the sampling range,and were analyzed using the Oldham-Parry analysis with the back reaction correction. The current-time transients were also analyzed using nonlinear regres- sion to determine the rate constant kf at each potential. The error in the resulting values of k are shown in Table 4.5 for all three f sets of data. The errors in the rate constants derived by the chronoamperometric analysis appear to be the largest, averaging around 402, while the rate constants derived from the 100 us polarograms are the most 102 Table 4.5. Comparison of Chronoamperometric and Pulse Polarographic Analysis of Transientsa with Finite Risetime.b Error in kf Derived by Chrono- Pulse Pulse E (mV) kf (cm/s) amperometryc Polarography 100 us Polarographylms 45.7 0.0411 +47.22 +3.02 +13.62 20.7 0.0668 25.9 -3.4 7.7 - 4.3 0.109 22.5 - 4.2 6.5 -29.3 0.177 30.7 - 0.9 8.5 -54.3 0.288 67.7 + 9.6 16.1 -79.3 0.468 d 44.0 e -104.3 0.762 d e e a100 us sampling time, 1 ms time range. Estg I 0 mV. kstd I 0.1 cm/s, - 2 Cox I 1 mM, Cred z 0, Dox I Dred I l x 10 cm ls. bT I 20 us. c One parameter nonlinear regression. dAnalysis failed to yield reasonable values of rate constant. ei/i too large for meaningful results. lim 103 accurate. The reason for this behavior can be seen upon examination of the shapes of the pulse polarograms and the position of the ap- parent limiting current. Especially on the 100 us polarograms, the amount of error in the current increases up the wave, so that the values of i/i11m are fortuitously close to ideal. This compensa- tion will occur to some extent whenever a larger-than—correct value of the limiting current is used. Since the error in the limiting current was considerably smaller with the 1 ms polarograms, the ef- fect is not as striking, although the results are still better than the chronoamperometric analysis. The apparent success of the normal pulse polarographic analysis under conditions in which the data are clearly suffering from some nonideal effect must be viewed with caution. ‘It is not clear whether such an arbitrary method can be relied upon to yield reliable esti- mates of electrochemical rate parameters under laboratory conditions. The success hinges on selecting the "right" limiting current for the conditions involved, and there would seem to be no way to determine this value from the data alone. 4.7. Comparison of the Effect of Adsorption on Normal Pulse Polarog: raphy and Large-Step Chronoamperometry The adsorption of reactants in large potential-step experiments leads to a relatively large number of cases to be considered, as we need to be concerned with the adsorption of two species both before and after the step. Since these large-step experiments are generally carried out with a very small concentration of product in the solution, 104 its initial degree of adsorption may be neglected. It has been de- termined that the adsorption of product has only a minimal effect at the final potential, so that in this work only the adsorption of the reactant needs to be considered. Even with this limitation, there are three possible variations: the reactant may be adsorbed at the initial potential, the final po- tential, or at both potentials. As was seen for irreversible reactions 1 in Chapter 3, these variations lead to different shapes in the result- ing normal pulse polarograms. These effects are shown in Figure 4.4. s Again, the deviations in the shapes of the polarograms were es- sentially the same as those seen for irreversible reactions. In all three cases, the ideal limiting current was reached only one or two hundred millivolts after the peak. The largest deviation is seen when the reactant is adsorbed at both the initial and final potentials. Adsorption which occurs only at the final potential leads to currents which are too small on the rising part of the wave, while the opposite is true for adsorption only at the initial potential. These two curves become identical at the top of the wave in the peaked region and beyond. One interesting aspect of these systems is the behavior of the polarograms as the reversible limit is approached. Flanagan and Anson (6) have examined normal pulse polarograms under Henry's Law adsorp- tion conditions, but with a reversible electrode reaction and a model which considers the depletion of the reactant molecules from the vicinity of the growing electrode due to the adsorption process. They observed waves which were too small, but otherwise normal, when 105 400 15 I II I 300 " _ 3 4.: 200b q d 0 H H 9 U 100 - -4 0 l l l l, 0 I200 I400 Potential (mV) Figure 4.4. Pulse polarograms illustrating the effects of reactant adsorption, with kstd I 0.1 cm/s, a I 0.5, Est 5 d I 0 mV, cm2/s, 100 us sampling time. c IlmM,D ID -1x10" OX ox red Curve 1: ideal system. Curve 2: Kix I KOx I 2x10-5cm. Curve 3: K1 I 2x10.S cm, K I 0. Curve 4: K1 I 0, K I 2x10-55 cm. ox ox ox ox 106 reactant and product were equally adsorbed throughout the experiment, which they attributed to reactant depletion. They also observed maxima and a shift in the potential range of the wave with differing degrees of adsorption for the reactant and the product. Because the adsorption coefficients and concentrations they chose lead to surface coverages equivalent to about 5 - 50 monolayers, it is doubtful that the Henry isotherm would apply. However, if their adsorption parameters are used in the simulation programs employed in this work, maxima at least 10 times larger than they report appear in the wave, which could be due to the depletion phenomenon. Other charac- teristics seem identical, except for the absence of the depletion effects. As an example of this effect, Figure 4.5 displays two polarograms with equal degrees of adsorption.~ One has a standard rate constant of 0.1 cm/s, while the other system has kstd I 3 cm/s. For comparison, Figure 3.4a illustrates the effect of the same amount of adsorption on a.totally irreversible wave. At 3 cm/s, there is no evidence of any deviation due to the nonideality. This is probably because the reaction proceeds so fast that all the adsorbed species reacts at the very beginning of the experiment, so that all the current observed, even at 100 us, is due only to diffusing species, as the theory pre- dicts. Since the limiting current is readily accessible from the nonideal polarograms, it is possible to derive rate constants from the data without the ambiguities involved in the polarograms distorted by finite potentiostat risetime effects. Again, a series of transients 107 300 " " 200 ' \ Current (uA) 100 F- '- 0 I L 1 1 0 -200 I400 Potential (mV) Figure 4.5. Pulse polarograms illustrating the effects of reactant adsorption at the onset of reversibility, with a I 0.5, I 1x101? cmZ/s, K1 I K I E . 0 mV, Cox - 1 “M’ Dox - Dred ox ox std 2x10.5 cm, 100 pa sampling time. Curve 1: kstd I 3 cm/s. Curve 2: kstd I 0.1 cm/s. 108 was generated for all three adsorption schemes out of which polaro- grams were constructed. The results of the analyses of both the original transients and of the polarograms are given in Table 4.6. Here, the difference between the analysis methods is not as distinct as was seen previously. The chronoamperometric analysis yields somewhat more accurate rate constants in almost every case. The overall error in the rate constants is quite large, regardless of analysis method used. 4.8. Implications for the Use of Normal Pulse Polarquaphy Because of the extensive use of normal pulse polarography in electrode kinetics studies as well as analytical work (53), some comI ments about the implications of the results of this work will be made. The normal pulse polarographic mode of large potential-step ex- periments is quite useful for diagnosing nonideal conditions. The presence of a peak resembling d.c. polarographic maxima is a clear indication of something nonideal in the experiment; the theory des- cribing the ideal experiment predicts no such shape. Furthermore, the exact shape of the maximum yields information on whether chemical or instrumental nonidealities are at fault. Peaks due to adsorption fall off rapidly to yield a constant, diffusion-limited value, while those due to finite potentiostat risetime show an extended region past the peak in which the current constantly decays. This diagnostic information is all that can be extracted from obviously nonideal polarograms; it is clear that data on the rising part, and even at 109 . ufiflumaOU one» any mo oaHm> oHnmcommou m uHlo cu noHHmw mHthmnd m .muHommH HomwaHomma How owHoH oou BHHH\Hw .aonmouwou unocHHsos nous nausea once .80 mloH x N I o sou u comm u smut .ao n|OH x N u ovum u sou .ao HHIOH a H u ewes n ammo .ae HHuoH a H - sees u see .au nuoH x H u e we 1 amen .e\~ae n.oH x H u seen u .o a mono .25 H u now .>5 0 I comm .m\ao H.o I name .owsmu oaHu me H .oaHu wcHHeaom m: OOHm m m H w m m.~m w m H «65.0 m.eoHu m w H m m.H- H.eH+ m 48H HeH eee.o m.aeu com mHH mmH w.mm v.5 I H.HI n.om N.mm 0.0m ww~.o m.em| «NH H.am «.mm H.om m.HHn m.m| o.mm o.mc o.mm- NNH.o m.m~u H.Hw m.nw n.0m ~.eH o.o~u «.ml e.~m «.mo m.~m aoH.o m.e I o.mm ~.oo m.ee H.nH m.mHn m.w| w.em m.eo c.mm mooc.o n.o~ nomm+ NHHH+ NH~H+ NH.a~+ N0.MHI Mm.m| N©.me+ Nm.m~+ NH.o~+ HHeo.o “.me .E am 66 .E .E 63 .E .3 «3 333 we 95 m as H 6: ooH as H e: ooH as H 6: ooH ououw< + ouowom uhHeo Houm< ehHmo muowom "noouomu< uonooum + unmuomum sons we eH uouum . .aoHumuomnd usuuomum Hows euHB mmuomHmcmuH mo mommHos< oHenuuwouoHom omHnm one UHHuoBouoaanooouno mo somHquEou .o.e oHnoa 110 the foot of the wave, are affected by the nonideal conditions. The implications of this work for electrode kinetics experiments have for the most part already been outlined in this chapter and in the preceding one. The main interest in these experiments is in the rising part of the wave (although errors in the limiting current will influence the results also). In a crude and not particularly general way, pulse polarography might be useful for distinguishing fast kinetically controlled re- actions from those which are reversible (or nearly so), since the maximum is reduced in size as reversibility is approached. The ab- sence of a peak (assuming that all is well with the instrumentation) can.be interpreted either as indicating a reversible reaction or simply that there is no reactant adsorption. This limited diagnos- tic ability might occasionally prove useful, however, in those cases when a reactant is known to be adsorbed at the electrode surface. The need for a reliable, fast-rise potentiostat in the study of rapid heterogeneous electron transfer-kinetics is especially obvious when one attempts to sample the current at short times. A general rule for the study of fast reactions might be that the risetime of the potentiostat be at least 100 times faster than the time at which the current is sampled. Diffuse-layer adsorption should not pose a problem if the experi- ments are carried out in l M supporting electrolyte at such a poten- Is.“ -—.——-—. tial that the total charge on the electrode is small, or if the samr «“— pling time is at least long enough that maxima are not visible along the wave. There could, however, be serious problems in systems with 111 lower supporting electrolyte concentrations, or in solutions in most nonaqueous solvents due to unfavorable double-layer conditions (en- hanced diffuse-layer adsorption), or increased solution resistance. Even though the Henry isotherm is probably not valid for significant degrees of diffuse-layer adsorption, the results of this work suggest that any weak reactant adsorption has a strong effect on the shapes of the pulse polarograms sampled at short times. As long as one doesn't attempt to extract rate data from obviously nonideal polaro- grams, the errors will probably be small. This does, however, form ‘ummnw a limitation on the maximum accessible rate constant under a given set of conditions. The use of normal pulse polarography in chemical analysis has both different procedures and different goals than in electrode kin- etics. Here reactant concentrations are often quite small (<10-4 M), and the only part of the wave which is of interest is the diffusion- limited plateau. The optimization of sensitivity is usually achieved by decreasing the sampling time to increase the measured current (53). Because of this need to enhance the sensitivity of the experi- ment, the risetime of the potentiostat plays. a critical role. It must be fast enough to allow the limiting current to be measured ac- curately as soon as possible after the potential step. Thus, the potentiostat, and the system itself, place a lower limit on the sensitivity of the analysis. The Henry isotherm for diffuse-layer adsorption is valid at the trace reactant concentration level, even to adsorption coefficients 112 greater than 2 x 10.5 cm. Analytical systems often have low support- ing electrolyte concentrations, are sometimes in nonaqueous solvents, and involve somewhat uncharacterized systems. Diffuse-layer or specific adsorption might be quite strong under these conditions. A basic problem in these analytical experiments is that the full polarographic wave is rarely recorded in applications such as flow injection analysis (54) or chromatographic detection (55). A potential 3 is chosen which is assumed to be well into the diffusion-limited region, and all the measurements are done at that single potential. 3 If this value is far enough into the plateau, adsorption of the analyte I should not influence the experimental results, although instrumental problems may still occur. It is thus quite important to verify that one is indeed in a well-defined (if not ideal) region of the wave for every variation which is made in the chemical system. A final, positive point can be made regarding the use of normal pulse polarography in analytical applications. Since the nonidealities studied here show very little dependence on the concentration of the reactant, working curves made from carefully made standards should be valid. As long as all conditions leading to the assorted non- idealities remain constant, accurate analyses might still be made, even in the presence of nonideal conditions. The next two chapters contain a study of two other methods used in the investigation of electrode kinetics. These small-step per- turbation techniques will be studied in much the same way as was done for the small potential-step experiments. The results of the three studies will be compared in an effort to determine the most reliable 113 method for the study of fast electrode reactions in the face of in- evitable experimental nonidealities. CHAPTER 5 COULOSTATICS EXPERIMENTS 114 ‘43qu '- 5.1. Description of Method The small-step coulostatic experiment, as it is usually practiced, requires both the oxidized and reduced form of a redox couple in soluI tion together with an electrode at such a potential that the system is at equilibrium. An injection of charge takes place (usually in the form of a short current pulse (23) or the discharge of a capaci- T“”[."_,'. tor) which causes the potential to increase by several millivolts. The system is no longer in equilibrium, as the Nernst equation demands a change in the ratio of oxidized to reduced species concentrations at the surface. The electron-transfer reaction begins in order to adjust this ratio, using electrons which are part of the injected charge. This causes a concentration gradient to be established in the solution near the electrode. Thus, charge leaks off into the solution and the overpotential decays back to the original equilibrium value. The rate of this decay is controlled by both the rate of the electron—transfer reaction, and by the rate at which molecules can diffuse through the solution. A general equation which describes the coulostatic overpotential- time transient has been derived by Keller and Kirowa-Eisner (56), although much earlier Delahay (12) and Reinmuth (13) both derived a more limited case of this equation. For a process Ox + ne- Red (5.1) OF+1H1W 115 116 under both activation and diffusion control, the overpotential-time transient is described by the following equation: ° 2 1, 2 11 n ' §E§{Y83P(B t)erfc(8t )-Bexp(y t)erfc(yt )] (5.2) where 31 Ta 1 Ta 1, B- +— -—-—1) (5.3) Zr. .3 41. T1 1 Ta 5 y--—--—— -———-1) (5.4) 21C 3 41¢ n° - 0163/(°11°A) (5.5) The charge transfer time constant Tc and the diffusion time constant Td are defined as follows: 2 2 l-o 0 Tc RICd2/(n F kstdcox cred) (5'6) 2 1,5 + 1,5 )] (5.7) ox ox Cred red It is traditional to identify two limiting cases of Equation 5.2. If Td << Tc, the overpotential decay is totally charge-transfer controlled, and Equation 5.2 reduces to 117 n I no exp(-t/Tc) (5.8) At the other extreme, when T >> Tc’ the rate of reactant diffusion d controls the overpotential decay, and Equation 5.2 reduces to n = no exv(t/rd)erfC(t8/Tdk) (5.9) Unfortunately, neither of these limiting cases is particularly useful to the electrode kineticist. The latter situation provides “-quc’!‘ A . ._. n no information about the kinetics of the redox couple under study, while the former is useful only for relatively slow reactions (k < 0.1) or at concentrations which are too high (>10 mM) to avoid std reactant ion migration or disturbances of the double layer. The above equations assume an instantaneous injection of the re- quired charge at the start of the experiment. This work will assume that the charge is applied in the form of a current pulse of large amplitude and very short duration. This model was chosen over the capacitance discharge method because of its more well-defined nature. Experimentally feasible pulse widths range from about 30 us to 500 us, the longer times being required for solutions of high resistance. As only a relatively small number of electrons are being used in this experiment, the presence of additional molecules of reactant at the electrode surface might be expected to have a significant effect on the coulostatic decay transient, considerably more so than a potential-step experiment, for example, in which a virtually unlimited number of electrons are available. Thus, even very weak reactant 118 adsorption could cause major deviations from ideality in the shape of the transient. 5.2. Analysis of Data from Coulostatics Experiments A number of methods have been used to extract kinetics and/or capacitance information from coulostatic overpotential decay curves. e.-;fl Originally, experiments were designed in such a way that simple charge- .‘-_A 1 I transfer control applied (12,13). Under these conditions, a plot of la n 3g, time has an intercept which is inversely proportional to the double-layer capacitance, and a slope which is directly proportional to the standard rate constant. Deviations from linearity due to an increasing contribution from diffusion increase with time, leading to curvature in the simple plot. ‘In this case, the initial slope of the curve would be used. Recently, nonlinear regression on the full decay equation (Equa- tion 5.2) has been used to provide estimates of the double-layer capacitance of the electrode and of the standard rate constant of the redox couple under study (57). This procedure has the advantage of extracting rate data from the experimental transient while the decay curve contains a significant contribution from diffusion. A comparison of coulostatic data analyses has been published by Kudirka, Daum, and Enke (4) which indicates that nonlinear regression is a superior technique for extracting charge transfer information . for experiments in which the ratio ‘rc/Td was less than about 10. Keller and Kirowa-Eisner (2) have analyzed the errors inherent in both of the above-mentioned procedures, and have determined the 119 optimal parameters for the determination of the capacitance and/or the standard rate constant. The study suggests that the best accuracy for the determination of the kinetics parameter can be obtained by analyzing data over a time range of twice the charge transfer time constant from the start of the experiment, for experiments with Tc/Td greater than about 0.5. Additionally, an analysis procedure based upon a transformation of the experimental data to the impedance plane and the use of the Laplace transform of Equation 5.2 has been suggested (58). v... rum. .‘1 In the present work, all analyses were performed via nonlinear regression on Equation 5.2 which, although somewhat time consuming, should give valid results for most values of the ratio Tc/Td. How- ever, if a transient is totally diffusion controlled, no kinetics information is available and the analysis (indeed, any analysis) will fail to yield a reasonable value for the standard rate constant. The optimal time range for the derivation of heterogeneous rate constants using nonlinear regression suggested by Keller and Kirowa- Eisner (2) was used in this work. This interval is equal to twice the charge transfer time constant, which turns out to be an experi- mentally reasonable window for standard rate constants less than about 5 cm/s. Twenty evenly spaced data points were used for the analysis of the nonideal coulostatics experiments. 5.3. Unique Aspects of Simulation The simulation of an ideal coulostatic experiment differs from that of the previously discussed potentiostatic experiment. The 120 overpotential varies with time, being a function of the flux of electrons at the electrode surface. In other respects, however, the potential has the same effect on the flux itself (through the rate constants) as it has in the previous techniques. The initial overpotential depends on the capacitance of the elec- trode and the amount of injected charge, as indicated in Equation 5.5. This is calculated as an initial condition in the ideal simula- tion. As electrons are transferred across the surface, the rate of decay of overpotential must be given by dn/dt . -F¢ (5.10) far/C62 Once the flux has been calculated in the usual manner, the equation can be applied in discrete form to calculate the change in the over- potential during that time increment At. The next calculation of the boundary conditions proceeds from this new overpotential. In the simulation of current impulse charge injection, the initial boundary conditions were calculated during a pre-time zero simulation period. Before this time, the concentration profiles are flat. Charge starts to be injected at a constant rate (112;: the current is switched on), the overpotential increases, and faradaic processes occur which distort the concentration gradient. At the end of the current pulse, the overpotential is less than that predicted by Equation 5.5, and concentration gradients have been established in the bulk of the solu- tion. This time is defined as time zero. During the charging period, the cha of sur cur so s11. tea: to ‘- effE 121 the overpotential increases as follows: dn/dt I Pam p/(Cd£'A) - F¢farlcd2 (5.11) The duration of the charging period and the current pulse amplitude Pamp determine the total amount of injected charge, Qinj' 5.4. Effect of Finite Charge Injection Time Current impulse charge injection can be thought of as a linear charging of the electrode double layer, during which, ideally, none of this charge has time to "leak off" due to faradaic processes at the surface and the diffusion profile is undisturbed at the end of the current pulse. In an actual experiment charge does leak off, so the charging is not quite linear and the experiment starts at a slightly smaller overpotential and with a concentration gradient al- ready set up in the solution. Intuitively, this would seem to lead to an apparent value of the capacitance which is too high, but the effect on the standard rate constant obtained from the overpotential- time decay curve is not obvious. The shape of the deviation produced by a finite charge injection time is shown in Figure 5.1. This shape is typical for a range of rate constants and conditions. The time it takes to apply a given amount of charge depends on a number of factors - among them are the pulse generator charac- teristics and cell solution resistance. Experimentally feasible in- jection times for such small amounts of charge as are required (on 122 Overpotential (my) I l I 1 0 5 10 Time (us) Figure 5.1. Coulostatic transients illustrating the effects of finite charge injection time, with kstd I 1 cm/s, Cd1 I 20 uF/cmz, 2 Qinj I 1 nC, area 0.02 cm., cox I cred 1 mM, Dox Dred 1x10-55 cmzls. Curve 1: ideal charge injection. Curve 2: tinj I 500 ns. 123 the order of 10.3 uCoul) for these experiments range from about 10 ns to 200 ns. All transients in this section were generated with injec- tion times in this range, and with standard rate constants between 0.1 and 10 cm/sec. Table 5.1 displays the error in the standard rate constants ob- tained for a series of analyses of nonideal transients with various concentrations of oxidized and reduced species, as well as the values of Tc and TC/Td for each case. These data are difficult to interpret due to a number of complica- tions. First, as the rate constant gets smaller, the time range of the experiment expands to maintain a 21c interval. Thus, for a given injection time, one is extracting rate data further and further from the nonideality. The observed effects might be expected to be smaller under these circumstances for this reason alone. Secondly, as the rate constants increase at a given set of concentrations, the ratio Tc/Td decreases. The smaller Tc/Td is, the more effect a small variation in the transient has on the derived value of the rate constant. Finally, one can observe that negative deviations are found in some of the cases studied, and positive deviations in others, $352) the derived rate constants are too high. Further, this seems to be a function of the ratio Tc/Td. Negative deviations in the value of the rate constant are observed when this ratio is less than unity, while positive deviations occur at values of rc/Td greater than one. This last somewhat curious result prompted some additional experi- ments under conditions that Tc/Td I 1. The results, also in Table 5.1, show that the correct standard rate constant is derived regardless Scleterfl H.“ a H a as . so «0.0 I swam . ao\m: 0N I «no .Hooo: N N 0H MI. 124 n naHo mm\~ao 0:0H x H I 00H: I Noam .mHthmnm nonmouwmu HmuaHHson Aaoo .0unxv Houoamumm omen 5.H 0.0 0.0 :::: ::: .50 «0.0 H.0 0.0 0.~ N.H :::: ::: 5.0H 55.H m.0 «.mH 5.0 «.0 N.H ::: 5.0 ~m0.0 H «.0H 0.0 0.0 + 0.~ + 0.0+ 5n.H 55H.0 m :::: :::: n.0H: «.0 : 0.0: 50.0 m00.0 0H 0H 0H 0.0 «.0 :::: :::: :::: Hm.0 0.0H H.0 0.0 + «.0 + H.0 + :::I :I:: «0.H H0.m 0.0 m.m : 0.H : 0.H : 0.0 : :::: H00.0 00.H H 0.5m: 0.0H: 5.0H: 0.0 : 0.~: <0H.0 00.0 m H 0H H.0 00.0 00.0 :::: :::: 5.0 ~.mm H.0 «.0 + 50.0 + no.0 + :::: :::: 50.H 5.5H 0.0 0 0 0 0 :::: 0.H 50.0 ~.H : 0.0 : m.0 : :::: :::: 50.0 mm.m H N0.~H: m.5 : 0.m : 5.H : :::: 50H.0 55.H m :::: Nm.0e: N0.0~: N0.nH: Nm.0: 500.0 00.0 0H H H 8H 2: on S oH ates. 0.3 6e 0.38 9.5 03 name mono Koo Amnv oaHH =0HuoofioH .osHH ooHuooncH ouHch on on: muooncuHH oHumumoHnoo scum moo>HHoa museumsoo some :H Houum .H.n oHan wk ac V0: hig dou the lee: able ment but 125 of the injection time. The reasons for this result are unknown, but it seems that there must be some sort of compensation effect in the devia- tion in different parts of the transient. It is probably true that the 2Tc time range over which the data is analyzed is significant. Other results indicate that correct rate constants are derived regard- less of the concentration of each species, or the ratio of the concen- trations. V A summary of the results obtained for experiments with various in- jection times is given in Figure 5.2, which shows the relative error in the derived value of kstd as a function of the ratio Tc/Td for given values of the ratio tinj/Tc' As one might expect, larger errors in the rate parameter are observed the larger the value of tinj/Tc (igg;, the closer the injection time is to the range of data which is analyzed). The minimization of deviations due to long injection time can be accomplished by attempting to adjust conditions so that the value of “re/1'd is close to unity. For large rate constants, however, this would necessitate increasing the reactant concentration (but not so high that the ions start making a substantial contribution to the double layer). Unfortunately, as the concentration is increased, the charge transfer time constant decreases, and data must be col- lected over a much smaller time range. Again, experimentally attain- able accuracy is determined by a compromise between ideal measure- ment conditions and physical practicality. ‘The value of the rate constant is needed to calculate Tc/Td but, of course, it is this parameter which the method is employed 126 10 l l T 4-9 - -: '0 U am a O b 2 - H u 1 ° & H H I!!! u I: .. .. 0 U H OJ Dr: -10 P- - l 1 l 0.1 1.0 10 1c / Td Figure 5.2. Relative error in standard rate constant vs. ratio Tc / Td' Curve 1: tin [Tc - 0.0188. Curve 2: j tinj/Tc - /‘tc - 0.133. Curve 4: t ITC - 0.188. 0.0565. Curve 3: t inj inj 0f tr 127 to measure. Any attempt to optimize an experiment in this manner probably will have to proceed through an iterative process, suc- cessively refining the estimate of the standard rate constant until consistent results are attained. It is not clear that this approach would not yield instances of false agreement, though. From the shape of the deviation from the ideal transient produced by finite injection time, one might expect values of the double-layer capacitance derived from nonideal transients to be too high. Table 5.2 shows the error in values of the double-layer capacitance obtained from the analyses which concurrently yielded the rate constant data in Table 5.1. The capacitance values are indeed too high. The data are summarized in Figure 5.3, which shows the relative error in the value of the capacitance as a function of the ratio Tc/Td at constant tinj/Tc' Obviously, the effect of finite charge injection time upon the apparent value of the capacitance is in general much smaller than was seen for the standard rate constant. For no value of Tc/Td is this error completely eliminated, however. 5.5. Effect of Weak Reactant Adsorption An excess of reactant at the surface of the electrode during a coulostatic experiment would be expected to produce a transient showb ing a steeper decay than would be seen if there were no adsorption. The enhanced surface concentration would lead to a greater reaction rate, and hence a faster discharge of the double layer. The extent of the deviation might be expected to be related to the rate of charge transfer compared to that of the diffusion process. For reactions .-..:~Lt («(5.0 {.oer—rd {.0 C..A— £7&:a~_~«mfi_~whp~t U!:-wUT~nd-~Avhv =50.»th -tfl>whma mootfiusowafiu kUXGQIEHQZGQ n: hehhm .N.W WNQRI .ao No.o I mmu< “Nau\m: oN I 0no 128 "Haoo: 0 0H I nch mm\Nau 0I0H x H coma I x000 .:0Hmmmuwmu HomaHHno: 5000 .cumxv umumamumnIozam 0.H 0.0 0.0 III III: 50 00.0 H.0 0.0 0.0 0.H II: III: 5.0H 55.H 0.0 H.0H 0.5 0.0 0.H III: 5.0 00.0 H 0.00 0.0H 0.0 0.0 H.0 50.H 55H.0 0 III II: 0.00 0.0H 0.0 50.0 000.0 0H 0H 0H 0.0 0.0 III II: III: H0.0 0.0H H.0 0.H 5.0 0.0 III III: 00.H H0.0 0.0 0.0 0.0 0.H 0.0 III: H00.0 00.H H 0.0 5.0 0.0 H.H 0.0 00H.0 00.0 0 H 0H 0.0 H.0 00.0 III III: 5.0 0.00 H 0.0 0.0 H.0 III III: 50.H 5.5H 0 0.0 0.0 0.0 H.0 III: H 50.0 0.H 0.0 0.0 II: III: 50.0 00.0 H N0.~+ 0.H 0.0 0.0 III: 50H.0 55.H I 0 III: N5.0+ NH.N+ N0.0+ N0.0+ 500.0 00.0 0H H H ecu cod on oN as ap\up Anne up Am\aov Azav Azav cums vmuu #00 Away maHH GOHuumficH .maHH 60H» IomhaH ouHch Cu man mucmwmcmue oHumumOHsoo Scum mvm>Hum0 mmucmuHommmu umkmAImHnson GH uouum .~.0 anma 0 129 8 1 I I 6 h- “, F, 4 “U U c: 'H .. N 34 u -' 3 HI U 8 U H O 94 2:- Q‘ 1" 2 o I L l 0.1 1.0 10 T / I Figure 5.3. ratio Tc / T 0.0565. Curve 3: t [Tc - 0.133. Curve 4: t Relative error in double-layer capacitance vs. d' Curve 1: tinj{Tc - 0.0188. Curve 2: /Tc - 0.188. inj inj tinj/Tc . w’ p... LIA camera“-- q 130 with small rate constants, there should be very little deviation from the pure charge-transfer control, Equation 5.8. However, the appar- ent time constant would be smaller than that calculated on the basis of bulk reactant concentrations. Once the influence of diffusion is present, the decay will obey neither Equation 5.8 nor Equation 5.2, which takes both charge transfer and diffusion into account. The deviations produced by a surface excess equivalent to about 102 coverage are considerably larger than those seen for finite charge injection time conditions. Figure 5.4 shows a family of coulostatic transients which were generated with a standard rate constant of l cm/sec and identical adsorption coefficients for the oxidized and reduced species. Note that significant deviations are seen for ex- tremely small surface excesses (0.5%.surface coverage) for the moder- ately fast (1 cm/sec) experiments. Systems which have differing adsorption coefficients for the oxidized and reduced species show deviations in the transients which are intermediate between the no-adsorption and equal-adsorption cases. Only a slight difference is seen for K - 0 cm, K - 2 x 10-5 ox red em and K - 2 x 10.5 cm, X - 0 cm. ox red The use of Equation 5.2 in the analysis implies that, for given reactant concentrations, a transient can never decay faster than the diffusion limited rate. Some of the transients which were generated for fast reaction rates and high amounts of adsorption appeared to the analysis to be decaying faster than should be possible; the results obtained therefrom are meaningless (and easily identifiable as such, since the k8t values are generally in the hundreds; the final result d 131 Overpotential (mV) I l l l 0 5 10 Time (us) Figure 5.4. Coulostatic transients illustrating the effects of 2 reactant adsorption, with kstd 1 cm/s, Cd1 - 20 uF/cm , 2 Qinj - 1 nC, area - 0.02 cm., Cox Cred 1 mM, Dox Dred 5 1x10- cm2/s. Curve 1: ideal system. Curve 2: Kox - K - red 6 leo‘6 cm. Curve 3: x -x - 5x10- cm. ox red deg prc ini con diz In ref rel eff: err: is ‘- fic: Val: 0f c the ofl Std resu adSO. in P: 132 depends on the termination routine in the nonlinear regression program). This problem is not unique to the nonlinear regression analysis; a log n vs, time curve will show no linear region, and its initial slope will be meaningless. Table 5.3 displays the error in the values of standard rate constants derived from transients for various concentrations of oxi- dized and reduced species, and with various values of K0; and Kred' In all cases, Kox - Kred' The values of Tc and Tc/Td are given for reference. Perhaps the most striking trend visible in these data is that the relative error in kstd is almost constant for a given adsorption co- efficient and rate constant, as long as C - C The relative . ox red' error increases somewhat as the concentration decreases. Also, there is a larger effect on the rate constant for larger adsorption coef- ficients, as would be expected. The fact that the relative error in kstd value of the adsorption coefficient indicates that it is the ratio is so dependent on the of surface excess to bulk concentration that is operative, and not the absolute magnitude of the surface excess. For example, a coverage of 102 with bulk concentration of 10 mM produces the same error in k as does a coverage of only 0.1% when C- - C - 0.1 mM. This std ox red result is somewhat disturbing for it indicates that decreasing the adsorption by decreasing the bulk concentration will be ineffective in producing a more reliable value of the standard rate constant. The slight increase in the relative error as the concentrations are decreased is probably due to the value of the ratio Tc/Td becoming .EOfithCm :_.< IZZIZEEE x023 30 0:: mucwumcth UHuwumcHzcu Souk at umzou 00%: :H houhm 0 «.3;qu 0.0.2.: 01% Qwéfih 133 .mus> HsmwcHawma m 0HmH> cu 0mHme mHthmqHuo0 mucmumaoo was: a« uouum .0.0 oHan 134 more unfavorable, so that the same deviations in a transient produce a larger and larger uncertainty in the derived value of the kinetic parameter. The above generalizations do not include the case when Cox ¥ C As can be seen in Table 5.3, the error in kstd is substantially red' higher when Cox - 10 Cred’ even though the values of Tc/Td are almost the same as those for Cox - C - 1 mu. A representative adsorp- red tion coefficient of 10"6 cm and standard rate constant of 1 cm/sec were used with various bulk concentrations Cox and Cred to generate a series of transients which were analyzed to yield values of kstd' These results are shown in Table 5.4, and one can note several points of interest. First, the minimum error in kstd is found when Co - C As x red' the ratio Cox/Cre deviates from unity in either direction, the rela- d tive error in kstd increases, but not symmetrically. This could be a consequence of going further from the standard potential and de- creasing the sensitivity of the experiment to the rate constant. Finally, the trend in kstd does not correlate well with Tc, Tc/Td, Cox/Cred’ or the total amount of adsorbed species. It was also noted that the injection of charge of the opposite sign produced only minor variations in the derived rate constant, except at very small values of Tc/Td. Table 5.5 shows the results of analyses of some transients gen- erated using unequal adsorption coefficients, and representative values of the rate constant (0.3 cm/sec) and concentration (Cox - Cred - 1 mM). The data show a rough symmetry about the Kox - Kred diagonal, and it 135 OX red Table 5.4. Effect.of Varying Reactant Concentration on Error in Rate Constant Deriveda From Coulostatic Datab with Weak Reactant Adsorption.c Error in kstd c (mM) c (mM) (us) 1' /T no - -2 5 mV no . +2 5 mV ox red c c D ' ' 0.05 1 23.8 0.0191 +93% +1132 0.1 1 16.8 0.0491 65 --- 0.2 1 11.9 0.117 40 --- 0.5 1 7.53 0.295 28 28 1 l 5.32 0.470 26 26 2 1 3.76 0.590 27 27 5 1 2.38 0.583 37 --- 10 1 1.68 0.491 55 --- 20 l 1.19 0.381 100 100 l 0.05 23.8 0.0191 +1012 +1202 1 0.1 16.8 0.0491 68 --- l 0.2 11.9 0.117 41 -- l 0.5 7.53 0.295 28 --- 1 l 5.32 0.470 26 26 1 3.76 0.590 27 --- 1 5 2.38 0.583 37 -- 1 10 1.68 0.491 55 -- 1 20 1.19 0.381 100 100 aTwo-parameter nonlinear regression. b -5 2 _ -3 kstd - 1 cm/s,2 Dox = Dred l x210 cm /s, Qinj - 10 ucoul, Cdfi a 20 uF/cm ; Area - 0.02 cm . CK - x =- 1 x 10'6 cm. 136 Table 5.5. Effect of Varying Adsorption Coefficients on Error in Rate Constant Deriveda From Coulostatic Data. Kred (cm) KOx (cm) None 1 x 10.6 5 x 10-6 1 x 10"5 2 x 10-5 none 0 +2.72 +14% +27% +45% 1 x 10‘6 +3.0: 6.3 18 32 52 5 x 10"6 11 19 37 55 83 1 x 10'5 29 34 56 81 120 2 x 10'5 48 56 86 123 176 aTwo-parameter nonlinear regression. bk - 0 3 cm/s C - C - 1 mM D a D - 1 x 10-5 cm2/s std ° ’ ox red ’ ox red ’ 10‘3 ucoul, c - 20 uF/cmz, Area - 0.02 cmz. Qinj ' d2 137 was found that values on each side of the diagonal are swapped when charge of the opposite sign is injected. It can also be seen that the deviations produced in kStd do not depend entirely on the sum of the surface excesses. The interpretation of the above results is not obvious, and no wide-ranging generalizations will be attempted regarding the error in l the derived value of k8t under differing degrees of adsorption. No d attempt will be made to map out every possible combination of concen- 3 .‘n qr? _ ..‘ trations, adsorption coefficients, and rate constants; however, some less general conclusions can be made. The most obvious result is that the error in kstd gets larger as the amount of adsorption increases and as the rate constant itself increases. It was also seen that for a given set of adsorption condi- tions, the least error is found when Cox - Cred' These results in- dicate that a substantial amount of error in kstd is present for even a moderately fast reaction when very weak adsorption is present. Since diffuse-layer adsorption can be of this magnitude, one can ex- pect the technique to yield erroneous results even for non-specifically adsorbed reactants whenever the equilibrium potential is at a point such that there is a significant amount of charge on the electrode. Thus, the concentrations of the oxidized and reduced species probably should be adjusted to bring the equilibrium potential as close as pos- sible to the p.z.c., minimizing the adsorption, although at the ex- pense of sensitivity of the transient to the rate parameter. The transients which were used to compile Tables 5.3, 5.4, and 5.5 also yielded values for the double-layer capacitance. Tables 138 I 5.6, 5.7, and 5.8 display the error in these values in the same for- mat as was used for the three previous tables. In contrast to the rate constant results, the derived value of the double-layer capacitance is smaller than expected in all cases, although the magnitude of the relative error was smaller. Some other interesting differences can be noted. The clearest indication of the general behavior of the.resu1ts of the analyses F of these nonideal transients can be seen in Table 5.7, and that is the smaller the overall reactant concentration, the less error is ob- i served in the apparent value of the capacitance. The relative error L seems to be inversely related to the charge transfer time constant, although the results in Table 5.6 indicate that it is not due to this factor alone. Table 5.8 shows the effect of varying the adsorption coefficients. Again we see the rough symmetry in the amount of error about the Kox a Kred diagonal. It seems, then, that conditions for the Optimization of a coulos- tatic experiment for measurement of the double-layer capacitance are not the same as for an experiment in which the kinetics parameters are of prime interest. One must lower the concentrations as much as possible to achieve the most accurate estimate of the double-layer capacitance. Since one is generally interested in the capacitance as a function of the electrode potential, it is not practical to ad- just the relative concentrations of the reactants so that the equilib- rium potential is near the p.z.c. to eliminate diffuse-layer adsorp- tion. If the adsorption is too strong, the transient will decay 139 A .ooHo> oHamcomoou 0HmHh ou uoHHow mHthoc< . Bo 00.0 I omu< . 00 . 0cH 00H xo 0 0Eo\m:_00 I 0 H500: 0I0H I 0I0H x H I 0 I a .cowmmouwou noosHHco: Houmaauoalosso III: III: III: 0.0: 0 III: 50 00.0 H.0 III: III: III: H.H: 0.0: H.0: 5.0H 55.H 0.0 III: III: III: H.0: 0.H: 0.0: 5.0 000.0 H 0H 0H III: III: III: 0.0: H.0: :III 0.0 0.0H H.0 III: III: :III H.H: 0.0: 0 00.H H0.0 0.0 III: :III III: 0.0: 5.H: 0.0: 00.0 00.H H H 0H 0.0: 0.H: 0.0: H.0: 0 III: 5.0 0.00 H.0 H.0H: 0.0: 0.0: 0.0: H.0: 0 50.H 5.5H 0.0 o o 0.0H: 0.0: 00.0: NH.0I 50.0 00.0 H H H 00.0: 0.0: H.0: 0 0 III: 50.0 000 H.0 II: 00.0: 00.0: H.0: 0 0 50H.0 55H 0.0 II: o 0 00.0: 0 0 500.0 0.00 H H.0 H.0 o o Ioaxm Ioexe Ioaxm Ioexm Ioexa oexm ap\ 0 Anne 0 Am\a60 A250 A230 0 0 0 0 5 00m won No f , ,1 I - II: I Isis ii; I: I- : I x 0 0 ohm u xOM Aaov uamHOHmwoou EOHuauomv< 0.:00uauom0< unouoo Iom xooz cu mac muaonoouH oHuoumoHsoo Scum o0o>Huo0 mucouHomeoo uoaoAIoHnson :H uouum .0.0 mHnoa Table 5.7. Effect of Varying Reactant Concentration on Error in Double-Layer Capacitance Deriveda From Coulostatic Data with Weak Reactant Adsorption.c 140 b Error in Cd£ Cox (mM) Cred (mM) Tc (us) Tc/TD n°-2.5 mV n°-+2.5 mV 0.05 1 23.8 0.0191 -0.1% -o.1z 0.1 1 16.8 0.0491 -0.3 --- 0.2 1 11.9 0.117 -O.3 --- 0.5 1 7.53 0.295 -0.4 -0.4 1 5.32 0.470 -0.5 -0.5 l 3.76 0.590 -0.7 -0.7 5 l 2.38 0.583 -1.1 --- 10 1 1.68 0.491 -l.7 --- 20 1 1.19 0.381 -2.9 -2.8 1 0.05 23.8 0.0191 -0.l% -0.22 l 0.1 16.8 0.0491 -0.3 --- 1 0.2 11.9 0.117 -0.4 --- 1 0.5 7.53 0.295 -0.4 --- l 1 5.32 0.470 -0.5 -0.5 1 2 3.76 0.590 -0.6 --- l 10 1.68 0.491 —1.6 --- 1 20 1.19 0.381 -2.8 -2.9 aTwo-parameter nonlinear regression. bkstd - 1 cm/s, Cd2 = 20 uF/cmz, Dox - Dred a 1 x 10"5 cmz/s, Qinj - 10.—3 ucoul, Area - 0.02 cm2. CK -I< a-1x10‘6 cm. OX red 141 Table 5.8. Effect of Varying Adsorption Coefficients on Error in Double-Layer Capacitance Derived8 from Coulostatic Data.b Kred (cm) K o"6 o‘6 o‘5 ‘ ox (cm) none 1 x l 5 x l 1 x l 2 x 10 none 0 0 -0.BZ -2.12 -4.22 1 x 10"6 o -o.1z -o.9 -2.3 -4.5 5 x 10'6 o -o.9 -2.9 -3.9 -6.5 1 x 10"5 -2.32 -2 4 -4.0 -6 o -9.4 2 x 10'5 -4.3 -4.8 -6.8 -9.5 -13.1 aTwo-parameter nonlinear regression. b -5 2 kstd - 0.3 cm/s, Cox 8 Cred = 1 mM, Dox a Dred - 1 x 10 cm /s, 2 -3 2 Cdi - 20 uF/cm , Qinj 10 ucoul, Area 0.02 cm . a- 7"” 142 faster than the diffusion limited rate, especially at lower concen- trations, and the analysis will fail to give a meaningful value for k Under these circumstances it would be risky to trust a capaci- std' tance value estimated simultaneously with the rate constant. Thus, it seems impossible to obtain a reliable estimate of the double-layer capacitance in the presence of a weakly adsorbed electroactive species if the electron transfer is even moderately fast (>0.1 cm/sec). n... 'I \f 2,.“ " . _ CHAPTER 6 GALVANOSTATIC DOUBLE PULSE EXPERIMENTS 143 Jr...- 6.1. Description of Experiment The galvanostatic double pulse technique was developed (15,59) as an improvement over the single pulse method in that it allows a pre- charging of the double layer prior to the actual measurement of the transient signal. This is important because it assures that all the current which is flowing is used in the faradaic (electron-transfer) process. An initial, short, but relatively large amplitude current pulse is applied to the cell, followed immediately by a second pulse of smaller amplitude. Depending on the duration of the first pulse and the relative amplitudes of the two pulses, a minimum in the over- potential-time curve can be observed at some time after the first pulse. This minimum occurs because the first pulse not only charges the double layer, but also sets up a steeper concentration gradient than is required to maintain the flux from the second pulse. This excess reactant causes the potential to return towards the equilibrium value until the gradient adjusts to the new flux and the potential begins to increase again. To perform a galvanostatic double pulse experiment, the worker interactively adjusts the magnitude and duration of the first pulse, keeping the current in the second pulse constant. The pulses are adjusted so that the minimum in the overpotential-time curve falls exactly at the end of the first pulse. The value of the over- potential at this minimum as a function of the duration of the first 144 145 pulse is conventionally used to derive the rate data of interest. The equations which describe the ideal overpotential-time behavior after the first current pulse are somewhat more complicated than those in coulostatics because the concentration gradients both before and after the pulse must be considered. For small overpotentials (15), ilG(t) (11-12)G(t-t1) n '- W- (6.1) d2 Cd£(Y-B) C(x) "J%[exp(82x)erfc(8xk)‘+ 28(x/n)k - 1] - B -J%{exp(y2x)erfc(yxk) + 21((x/1r);5 - 1] (6.2) Y l-o 0 ~ C C B/Y" std 021: red( 115 +4.?) +/_ oxDox credDred 2 2 l-a 0 [wk (c1 “C °‘ )<——— + —‘-;;—)}2- n F kstdc“) F:m‘llz std ox Cred C D8 RTCdz ox ox CDred red (6.3) It is also possible to define the time constants of the system. The charge transfer time constant Tc and the diffusion time constant Td are given by the expressions 2F 2 c1- -oa Tc 8 RTCdl/(DF cox Credkstd) (6'4) 146 stdC ox Cred rd = II/[(—J;)—;§- + Tw cl'ac 0‘ 32 (6.5) ox ox CredD red The consequences of adjusting the experimental parameters, par- ticularly the current in the second pulse, are the key to the success of this method. One can optimize the experiment for the particular rate constant, concentrations, and double-layer capacitance simply by adjusting the pulses appropriately. This sort of "fine tuning" is not possible in the usual potentiostatics experiments; perhaps a double potential-step method would be useful for establishing a concentration gradient in the solution before the start of the experiment. 6.2. Analysis of Data from G.D.P. Experiments As stated above, the galvanostatic double pulse experiment is typically performed by adjusting the amplitude of the first pulse until the minimum in the overpotential-time curve occurs exactly at the beginning of the second pulse. This process is repeated, varying the duration of the first pulse, while keeping the amplitude of the second pulse constant. These minimum values of the overpotential can be plotted against the square root of the pulse time to yield an inter- cept which is inversely proportional to the-standard rate constant for the redox couple (15): 2 F2 Cl-OIOI o kstd . RT12/(nF Cox crednmin) (6'6) 147 This relationship is obtained (60) by differentiating Equation 6.1 with respect to time, and setting (dn/dt)tmin equal to zero. This yields an expression for the minimum overpotential as a function only of 12 and t1. Setting t1 equal to tmin’ expanding the exp(x2)erfc(x) function, and dropping all but the first term of the expansion leads to an expression in t? with the intercept given by Equation 6.6 above. Because there are some instrumental difficulties in observing the overpotential at the exact point at which the current is switched, Nagy (60) has developed equations which allow the minimum in over- potential to fall at some defined time after the first pulse ends. The result of this work is a set of alternative equations relating the intercept of the “min gs, t? plot to the standard rate constant. All of these methods involve approximations in the linearization of the overpotential minimum-pulse time data. The obvious alternative to the above procedures is simply to use nonlinear regression to fit the transient to the explicit equation, Equation 6.1. As with coulostatics, the analysis will adjust two parameters, the standard rate constant and the double-layer capacitance. In fact, Nagy (62) has shown that this type of curve-fitting is superior to the conven- tional analysis in the accuracy of the derived rate constants in the presence of random measurement errors. As long as a minimum is ob- servable, the advantage of precharging the double layer remains (61). Although no optimal time ranges for the nonlinear regression analysis of galvanostatic double pulse data have been discussed, Nagy (62) has established some guidelines for single pulse galvanostatic 148 experiments: t2 - 10 Tc if rd > 10 re (6.7) I:2 a rd if Tc < Td < 10 TC (6.8) t t2 - 35(Tc + rd) if Tc > Td (6.9) I These time ranges were used for all the following work.when analyzing .7 ‘wwaa~* *1 the g.d.p. data by nonlinear regression on Equation 6.1. The effect of finite galvanostat risetime has been considered in the literature (1), and an explicit equation has been derived assum- ing a linearly rising current pulse. Nagy (61) states that the non— linear regression analysis of g.d.p. data is quite sensitive to this risetime when the ideal equation (Equation 6.1) is used in the analysis. However, in his study of the effect of measurement precision, he found that the results which were obtained for the errors in the rate constants were essentially independent of the specific equation which was used in the regression. Finite measurement precision had the same effect on the results whether the galvanostat was considered ideal or not. Because galvanostat risetime has been previously discussed in the literature (1), it was not considered here. Given the sensitivity of coulostatics to weak reactant adsorption, however, it was of interest to investigate the effects of this chemical nonideality on the results of the g.d.p. experiments. To simplify the procedure, it was assumed that the galvanostat was indeed ideal. In view of the above-mentioned measurement precision study, this should yield 149 essentially equivalent results as would using a nonideal galvanostat model. 6.3. Unique Aspects of Simulation The digital simulations of this technique were performed in a manner similar to those in preceding chapters. The constant current condition was identical to the current impulse coulostatics charge injection period, where the change in overpotential with respect to time can be expressed dn/dt - 11/(Cd2'A) - N (6.10) far/Cd2 The constant current began at time zero, however, and was reduced to the smaller value at the appropriate time. (When the second current is zero, the experiment is identical to current impulse coulostatics.) Twenty points were recorded at equal intervals along the simu- lated transients over a time period specified by the above conditions (Equations 6.7-6.9). The parallel simulation scheme was used for all transients which were to undergo the nonlinear regression analysis. Conventional simulations were used for those transients subjected to the conventional g.d.p. analysis. In these few cases, the minimum values were recorded manually as the current parameters were adjusted interactively for the various pulse times. 150 6.4. Shape of Deviations Due to Reactant Adsorption The effect of weak reactant adsorption on the galvanostatic double pulse transient is shown in Figure 6.1 for a typical system with varying amounts of adsorption. There are two features of the overpotential-time curves which are immediately noticeable. The over- potential which is attained after the first pulse is lower than ex- pected and is decaying at a faster rate when adsorption is present. The reasons for this are exactly analogous to those for coulo- statics. There is more reactant at the surface when the species is adsorbed, so electrons are lost to the faradaic process faster. This keeps the electrode from charging as fast as it might, and the result is a lower overpotential, even though the number of electrons flowing into the cell is the same. There are no obvious features which distinguish these nonideal transients from those which are ideal. The minimum.is shifted, and the decay is steeper, but in the performance of an experiment these would be interpreted as a maladjusted current pulse sequence rather than any chemical nonideality. Indeed, after suitably adjusting the current in each pulse, it is not at all obvious that adsorption is present. The larger than ideal currents necessary to compensate for the ef- fects of weak reactant adsorption are used in the equation describ- ing the ideal transient (Equation 6.1), so one would expect that the rate constants derived on the basis of this equation might be in error, even if the absolute shape of the measured transient was identical to Hi WEN 151 3.0 l ' 208 P d E H II! '0'. _ q 1.: 5 o 2 G- H g '\ 206 - q 3 4 2.4 I I 0 10 20 Time (118) Figure 6.1. Galvanostatic double pulse transients illustrating - 0.3 cm/s, C - the effects of reactant adsorption, with kst dl d 20 uF/cmz, 11 - 0.055 A/cmz, c1 - 1 us, 12 - 0.002 A/cm?, -5 2 cox - Cred - 1 mM, Dox - Dred 1x10 cm ls. Curve 1. ideal system. Curve 2: K - K - 1x10-6cm. Curve 3: K - K ox red ox re 3x10-6cm. Curve 4: K - K - 1x10"5 cm. ox red d 152 the ideal case. Thus, any rate constants derived on the basis of this equation must be in error. It is not clear, however, how much error would be produced, or even whether the derived parameters would be too high or too low. 6.5. Initial Investigations Galvanostatic double pulse experiments are unique in that there are a large number of parameters which can be adjusted for each chemi- cal system. The pulse parameters control the shape of the resulting transient: The first pulse mainly controls the overall overpotential change at the beginning of the experiment, while the second pulse controls the position of the overpotential minimum and steepness of the measured transient. It must be determined whether the results of the nonlinear regres- sion analysis are dependent on the adjustable experimental parameters when weak adsorption is present. (If the experiment were ideal, of course, there would be no problem because the equations take these experimental parameters into account.) There are three adjustments which can conveniently be made by the experimenter in setting the pulse parameters: 1) the length of the first pulse, 2) the overall potential change during the first pulse, and 3) the position of the minimum along the transient. In order to determine the influence of these variations on the results of the nonlinear regression analyses, three sets of transients were generated under identical adsorption conditions. The errors in and Ext th: ‘10 ad £01 311a. 153 the values of k8 and Cdl derived from these transients are listed td in Table 6.1. The first group shows that the effect of varying the first pulse time is quite small, with slightly larger errors from longer pulse times. This is probably due to the reduced amount of kinetics information available at these longer times. The second and third groups in the table show that the overpotential range of the experiment and the position of the minimum along the transient influence the error in the rate constant or in the double-layer capacitance only to a very small degree. It is now possible to proceed with a systematic variation in the amount of adsorption to observe the errors in the derived rate constants and capacitances. In these simulations, the initial pulse time will be held at 1 us for convenience, while the currents will be adjusted to produce a minimum overpotential of about 3 (+/-0.5) mV at a position roughly 10 to 302 along the transient. These small variations should have only minor effects on the results. 6.6. Effect of Adsorption in Nonlinear Regression Analysis A series of transients was generated to investigate in a general way the effect of weak reactant adsorption upon the results of non- linear regression analyses. For this preliminary work, the concentra- tion of both the oxidized and reduced species as well as their dif- fusion and adsorption coefficients were assumed to be equal. The 6 cm to 2 x 10.5 cm adsorption coefficient was varied from 1 x 10- for a series of rate constants. The results of the nonlinear regression analyses of these transients are shown in Tables 6.2 and 6.3. 154 Table 6.1. Effect of Adjustable Experimental Parameters on Error in Rate Constant and Double-Layer Capacitance Deriveda From G.D.P. Transientsb with Weak Reactant Adsorption.c t1 (us) tmin (us) nmin (mV) Error in kstd Error in Cdfi 0.5 2.8 2.6 +69% -4.32 l 3.8 2.7 71 -4.3 1.5 2.7 2.5 72 -S.0 2 3.3 2.6 74 -5.3 3 4.7 2.6 77. -6.0 1 2.5 .48 +672 -4.42 l 2.5 .97 70 -4.6 1 2.5 1.94 70 -4.6 1 2.5 3.88 70 -4.6 l 2.5 7.28 71 -4.7 l 2.5 2.7 +71% -4.72 1 6.4 2.7 71 -4.6 1 11.4 2.5 71 -4.6 1 17.8 2.3 71 -4.6 aTwo parameter nonlinear regression. 2 kstd - 2.3 cm/s, Cdl - 20 uF/cm , Cox = Cred 1x10- : 1 mM, Dox a Dred g cmzls. 5 c I 1 x 10- cm. Kox . Kred 155 Table 6.2. Error in Rate Constants Derived8 from G.D.P. Transientsb Due to Weak Reactant Adsorption (K.ox - Kred)' kstd (cm/s) xox - Kred (cm) 0.1 0.3 1.0 3.0 1 x 10‘6 +22 +6.32 +24.sz +25.2z 3 x 10"6 6.3 19.8 125 c 1 x 10'5 22.3 70.1 c c 2 x 10‘5 45.9 135 c c aTwo parameter nonlinear regression. bt 2 l - 1 us, i1, i2 varying, Cdfi - 20 uF/cm , Cox - C - 1 mM, 5 red D - D - 1 x 10- cm2/s. re ox d cAnalysis failed to yield a meaningful value. 156 Table 6.3. Error in Double-Layer Capacitance Derived3 from G.D.P. Transientsb Due to Weak Reactant Adsorption (Kox . Kred)° kstd (cm/s) ox - xted (cm) 0.1 0.3 1.0 3.0 1 x 10‘6 +0.01: -o.12 -0.42 —2.82 3 x 10'6 -o.2 -0.7 -4.1 c 1 x 10"5 -2.3 -4.6 c c 2 x 10'5 -7.0 -9.3 c c a Two parameter nonlinear regression. b 2 t1 - 1 us, 11, 12 varying, Cdi - 20 uF/cm , C = C ox red - 1 mM, 5 D - D - 1 x 10- cm2/s. ox red cAnalysis failed to yield a meaningful value. 157 Table 6.2 shows the error in the rate constants which are derived from these nonideal transients. In all cases, the derived rate con- stants are too high. There were instances in which the overpotential decay was so steep that no meaningful value of either the rate con- stant or the double-layer capacitance could be derived. This prob- ably indicates that the curves decayed faster than diffusion control would allow under these conditions. It can be seen that only a very small amount of adsorption causes very serious errors in the rate constant at values of l cm/s or more. This extreme sensitivity to relatively minor deviations is due to the lack of kinetics information in the shape of the transient. At smaller rate constants, there is still a fairly large amount of error present, except when the adsorp- 6 cm). tion is very small (Kox - Kred - l x 10- The errors in the capacitance values which were derived simul- taneously with the rate constants are shown in Table 6.3. Here we see that the errors are considerably smaller than for the derived rate constants, but that they do approach 102 for large adsorption co- efficients. The probable reason for this is that, although there is very little kinetics information left as diffusion control is ap- proached, the transient is still quite sensitive to variations in the capacitance. The error that is produced is due to the apparently larger extrapolated overpotential at the end of the first pulse, which would imply a smaller capacitance. The tendency for this to occur increases as the redox reaction rate increases, so we observe more error in the capacitance at faster reaction rates. Even though some of the capacitance values seemed reasonable .'-" 1.1.1.2.:3 . E- ‘t'. in. 158 when the corresponding rate constants clearly were not meaningful, their values were not reported in Table 6.3. It is thought doubtful to trust one parameter when others derived simultaneously are obviously in error. Another series of transients was generated to examine the effect of varying the individual adsorption coefficients in a typical system (kstd - 0.3 cm/s). The resulting errors in the derived rate constants are shown in Table 6.4. As in coulostatics and chronoamperometry, we see that the same general rule applies: The more adsorption, whether of the oxidized or the reduced species, the greater the error in the rate constant . The slight asymmetry in the data is due to the inherent asym- metry of the experiment. Even though the electrode is initially at the equilibrium potential, the current flow causes one half of the redox couple to be a reactant, while the other becomes the product. Thus we see that the adsorption of the reactant has a slightly larger effect on the shape of the transient than does adsorption of the product. The asymmetry in Table 6.4 has been found to be reversed if the current flows in the opposite direction. The trends in the errors in the double-layer capacitance values are the same as those for the rate constants, with a maximum error of about 10% when the adsorption coefficient of each species is the same, and only about 32 when only one of the species is adsorbed. Finally, it is of interest to study the effect of varying the concentration of the reactants. It was determined that it is the value of the adsorption coefficient which determines the error, and 159 Table 6.4. Error in Rate Constant Deriveda from G.D.P. Transientsb Due to Weak Reactant Adsorption (Kox i Kred)° Kred (cm) ox (cm) 0 1 x 10'6 3 x 10'6 1 x 10'5 2 x 10‘5 0 ---- +2.82 +8.32 +23.72 +36.42 1 x 10"6 +3.12 6.3 12.2 29.0 43.2 3 x 10‘6 9.5 13.0 19.8 39.3 56.5 1 x 10‘5 27.0 32.3 42.1 70.1 96.1 2 x 10'5 42.6 49.2 61.9 99.7 135 a Two parameter nonlinear regression. b 2 5 20 uF/cmz, c - c . 1 mM, D - 0 - 1 x 10' cmZ/s. 0x re 0x d red 160 not the absolute magnitude of the surface excess when there is equal concentrations of both species in the solution. Table 6.5 shows the error in both the standard rate constant and the capacitance derived from transients generated while varying the concentration of one half of the redox couple. It can be seen that equal concentrations of the oxidized and reduced species produces the minimum error, with the error increasing as the ratio of the concenr trations deviates from unity in either direction. The error in the double-layer capacitance, on the other hand, is smallest at the small- .4— ‘2' .- a. ‘ a est concentrations. As was the case with coulostatics, it is not possible to correlate these effects to the experimental variables in any systematic way; the exact amount of error depends on too many factors for correlations to be particularly useful. Some general conclusions can be made, however. The uncertainty in the rate constants deriVed using nonlinear regres- sion from g.d.p. transients is minimized when the equilibrium.poten- tial is the formal potential (1:3;J when the concentration of the two species are equal). Attempting to decrease the diffuse-layer adsorp- tion by adjusting the system for a new equilibrium potential closer to the p.z.c. will only work if the adsorption can be made negligible; otherwise, the accuracy gained by the smaller degree of adsorption will be counterbalanced by changing the concentration ratio from unity. 6.7. Effect of Adsopption in Conventional Analysis Conventional simulations were performed for two systems to compare the performance of the nonlinear regression analysis with the 161 Table 6.5. Error in Rate Constants Derived8 from G.D.P. Transientsb c Due to Weak Reactant Adsorption (Cox # Cred)' ox (mM) Cred (mM) Error in kstd Error in Cdz 1 0.1 +5002 -2.72 l 0.3 110 -3.4 l l 70 -4.6 l 3 84 -7.7 1 10 150 -12. aTwo parameter nonlinear regression. b 2 t1 - 1 Us, 11 - 0.055 A/cm , 12 varying, kStd I 0.3 cm/s, Cdl 2 -5 20 uF/cm , Dox Dred l x 10 = l x 10-5 cm. cm2/s. c K’ox - Kred e‘ con’ Ira 110» were botl refJ dig' 910 pt: pl: ce CC PE Ch: f0] 162 conventional ”min-zg' ta analysis. One of the systems chosen for this 1 treatment failed to provide any kinetics information under the none linear regression analysis, while the other yielded rate constants which were in error by about 702. Figures 6.2 and 6.3 show the nm vs. tk plots for each system in.-- 1 both with and without adsorption. The ideal case was simulated for reference because the simulated data were generated using conventional digital simulation routines. The linear regression lines are also plotted on the graph. It is clear that the presence of adsorption W‘WT'I“ ‘ “ " produces large deviations in both the slope and the intercept of these plots, as well as some degree of nonlinearity in the data. The inter- cepts from the linear regression together with the corresponding rate constants derived from these values using Equation 6.6 and their ap- proximate uncertainties are shown in Table 6.6. These results are strikingly improved over those derived by the nonlinear regression procedure. ‘In the first system, where kstd - 0.3 cm/s, we see only a 32 error in the rate constant compared to about 702 from the other analysis. The other system, for which nonlinear regression failed entirely to yield a reliable rate constant, yields a result by this analysis which is in error by roughly 602. The surprisingly more accurate results afforded by the conventional analysis were seen for a wide range of rate constants and adsorption parameters. Apparently, the method of adjusting the current pulses to position the minimum at a reproducible time in the experiment is responsible for this improvement. During the first current pulse, not only does 163 Overpotential Minimum (mV) 2.5 I I I 0 l 2 t}: (1:38) Figure 6.2. Overpotential minimum vs. t? for conventional g.d.p. - 20 uF/cmz, i - 0.003 A/cmz, d d1 2 5 - C - 1 mM, D - D - 1x10- cmZ/s. Lines are least- ox red ox red analysis, with kst - 0.3 cm/s, C squares best fit. Curve 1: ideal system. Curve 2: Kox - Kred - 5 1x10- cm. 164 6 E 2 v4 :3 E 4 H d H «U a Q) U 0 a. H 221 2 o l l l o 1 2 t? (11815) Figure 6.3. Overpotential minimum vs. t? - 3 cm/s, i2 - 0.02 A/cmz, other parameters for conventional g.d.p. analysis, with kstd as in Figure 6.2. Lines are least-squares best fit. Curve 1: ideal system. Curve 2: K - K - 1x10.5 cm. ox red 165 Table 6.6. Results of Conventional Analysis of G.D.P. Data8 with Weak Reactant Adsorption.b Intercept (mV) Derived kstd (cm/s) kstd = 0.3 cm/s Ideal 2.673 0.299 t .001 Adsorptionb 2.588 0.308 1 .002 kstd a 3.0 cm/s Ideal 1.619 3.289 t .16 Adsorptionb 0.982 5.423 t .67 8C t C - 1 mM D a D = l x 10.5 cmzls not parallel ox red ’ ' ox red ’ simulation. bK - K - l x 10.5 cm. ox red 166 the double-layer charge to the appropriate value, but the faradaic reactions of bulk species and adsorption occur as well. The extrapo- lation back to time zero which is conventionally done to correct for this concentration polarization during the first pulse evidently also corrects for the reaction of the adsorbed species. The t? plot lin- earizes the contribution from the diffusing species, as seen in Figures 6.2 and 6.3 for the ideal system, but the adsorbed species distorts the diffusion profile, rendering curvature in the data. It can be seen that a rough, curved extrapolation yields an intercept even closer to the ideal value, and hence a more accurate rate constant. 6.8.v Ipplications for the Use of Small-Step Methods It was found that experimental nonidealities in small potential- step, coulostatics, and galvanostatic double pulse studies of fast reactions produce a large amount of error when the resulting tran- sients are analyzed with nonlinear regression. In the face of non- ideal conditions, the traditional advantage of nonlinear regression (i;g;, the sensitivity of the analysis to minor variations in the shape of the transient) actuallyworks against the derivation of ac- curate rate constants from experimental data. As the amount of ki- netics information in the transient decreases (for faster reactions), a deviation of a given magnitude will produce a greater error in the rate parameter. These results should serve as a warning to those who routinely use nonlinear regression analyses to reduce their data. The equation 167 to which the experimental data is fit must accurately reflect the processes which are occurring in the experiment. This is especially important when the curve contains only a small amount of the informa- tion of interest. (Of course, the nonideal data examined here could, in principle, be adequately fit by an equation which takes the non- idealities into account. In this case, nonlinear regression would probably be the analysis method of.choice.) The surprising success of the conventional galvanostatic double pulse analysis was indeed a welcome result in the midst of failing nonlinear regression analyses for every other experimental method examined. Ideally, of course, an equation might be derived to allow a valid extrapolation of the ”min gs, pulse time curve to zero time, but even the rough, linear approximation produces much more accurate results than those obtained by nonlinear regression analyses, and even yields estimates of the rate constant under conditions in which nonlinear regression was not able to do even that. Other methods are also used to study the rates of quasi-reversible reactions, in particular, a.c. polarography. This method was not examined in this study because of the lack of a closedrform solution to the overall current response to the small a.c. potential perturba- tion. Conventional simulations yielded responses which were too in- accurate to be of use. It would be expected, though, that adsorption of the reactants would interfere with the measurement of fast rate constants here as well. Another method sometimes used to study the rates of fast electrode reactions is cyclic voltammetry. This tech- nique was not examined because preliminary studies (with conventional 168 simulations) showed only very minor effects due to reactant adsorp- tion; indeed, the observed errors were less than other known errors (gflgL, iR drop) for these experiments. Thus, it seems that the galvanostatic double pulse experiments with the conventional data analysis will yield the best estimates of fast electron-transfer rate constants in the face of inevitable experi- mental nonidealities. W’“ CHAPTER 7 SUGGESTIONS FOR FURTHER RESEARCH 169 ac ra ad EV St pr Fr: tic Coq nee SQr Sit is £0]: to‘ app] the 7.1. Extension to Other Adsorppion Isotherms While this work has illustrated that the influence of weak reactant adsorption becomes pronounced only when one attempts to measure the rate constants of very fast reactions, it is possible that stronger adsorption will influence transient responses to a larger extent even when slower reactions are studied. There are two isotherms in relatively common use for describing strong adsorption (1): the Langmuir isotherm and the Frumkin isotherm. The Langmuir equation needs two parameters to describe the adsorption properties, as it takes into account a saturation-coverage limit. The Frumkin isotherm takes this one step further (and adds one more ad- justable parameter) in considering adsorbate-adsorbate interactions. Either of these isotherms can be used in more complicated simula- tions to study the influence of strong adsorption. The study is made considerably more complex than the Henry's law case in that we now need several parameters (and their dependence on potential) to de- scribe the adsorption properties of each species in the system. The situation is further complicated by the fact that specific adsorption is usually not in equilibrium over the time scale of these experiments; for a realistic model, adsorptionedesorption kinetics would also have to be considered. Despite these complications, it should be possible to apply some appropriate simplifying assumptions and obtain useful results from the simulation of these systems. A more realistic model could aid 170 171 considerably in the interpretation of experimental results, as described below. 7.2. Alternative Analysis Procedures As suggested in Chapter 3, these more realistic models could be applied in a nonlinear regression analysis of data which was recorded in such a way as to maximize the extent of the nonideal behavior. Not only could rate data be recovered, but also information about the adsorption properties of the reactant. Two approaches suggest them- selves. The study of systems with strong adsorption could yield separate kinetics information for the adsorbed reactant and the diffusing reactant. At present, it is difficult to separate the overall faradaic current into the various components, so that the kinetics of the ad- sorbed species must be studied under conditions which minimize the reaction of the diffusing species. Two methods are used for this: a very small bulk concentration of the reactant (which requires very strong adsorption), or very fast experiments so that very little re- actant has time to diffuse toward the surface. A.simulation-based analysis might be successful in determining the kinetics of both re- actions simultaneously or even the adsorption-desorption kinetics of the species in question. A second possibility for simulation-based analysis is the compen- sation for nonideal instrumentation. The actual applied waveform (potential-step, current-step, etc.) could be measured independently of the response transient and used in a simulation analysis. This 172 approach might also make it relatively easy to do any resistance comr pensation at the analysis stage instead of guessing before the experi- ment is performed. By actually measuring the applied waveform, the need for some instrumental model is eliminated, making the entire pro- cedure more reliable. Implicit in the above discussion.is the assumption that charging currents due to double-layer capacitance in potential-step experiments make no contribution to the analyzed transients.. It is hard to ensure this because of iR-drop problems, even though the current may not be sampled until after the applied potential is at the desired value. Although small-step coulostatics and g.d.p. experiments take the double- 1ayer capacitance into account, the theory assumes that it remains constant throughout the experiment, which may not be true in the face of specific adsorption of the reactants. Additionally, a change in the number of adsorbed ions over the course of the experiment could cause nonfaradaic current to flow if the electrosorption valency of the adsorbate was large enough to affect the charge on the electrode. These factors could be examined in more detail to see if they are indeed large enough to produce deviations in the shape of the trans sients. The flexibility of digital simulation coupled with the generality of the nonlinear regression analysis gives these procedures much potential. On a mainframe computer, these analyses could become as convenient as a linear least squares calculation. APPENDIX APPENDIX A SAMPLE SIMULATION PROGRAM The following program uses the parallel simulation method to pro- duce an error curve to be impressed on a calculated ideal small-step chronoamperometric transient (see Chapter 2). Although comments have been provided in the listing itself, a few explanatory notes will be given. The listing produced by FLECS includes indentation to show the level of the control structure of an individual statement. The vertical columns of dots allow the level to be traced from.page to page. The statement "FIN" serves only to indicate the end of a control level. In addition to the initialization and the parallel simulations, there are quite a few lines of code dedicated to a convenient user interface. The internal procedure DISPLAYAMODIFY-PARMS allows any number of the variables in the simulation to be adjusted during a session with a minimum of effort. The octal (base 8) constants (218;: "33) in WRITE statements to unit 7 (the terminal) are DEC VT-52 compatible escape sequences for controll- ing the position of the cursor, clearing the screen, etc. File output statements are not shown in this listing for clarity, al- though all regular programs include this provision. The extra state- ments include OPEN/CLOSE logic for new files, a data set header section which records all the simulation parameters, and output statements for the simulated data themselves. 173 0000000 000 174 FLECS/RT-ll V28.01 PAGE 1 PROGRAM OSIM GSIM - CHRONOAMPEROMETRY SIMULATION PROGRAM BY E. SCHINDLER: 5/82 * OUASI-REVERSIBLE KINETICS * LINEAR POTENTIAL RISE DIMENSION C(100):P(100):CNEH(100):A(14) DIMENSION CI(100):PI(100):DX(100) REAL KF:KD:KSTD:KFI:KBI BYTE ERR EGUIVALENCE (A(1): KSTD): (ACE): ESTEP) EGUIVALENCE (A(3): COX): (A(4): DOX) EGUIVALENCE (A(5): CRED): (A(6): DRED) EGUIVALENCE (A(7): ALPHA): (A(S): AREA) EGUIVALENCE (A(9): DXI): (A(10):OGQ) EGUIVALENCE (A(11): BETA): (A(12):DTOUT) EGUIVALENCE (A(13):TFINAL):(A(14):TRISE) DEFAULT PARAMETER VALUES DATA Al. 1: 0.: 1.E-6: 1.E-3: 1.E-6: 1 1.E"30 .5: .02: 5.5-'6: .1: .4: 1.8-'4: i 2.015-3: 1.5-SI 175 FLECS/RT-11 V28.01 PAGE 2 INITIALIZATION SECTION WHILE (.TRUE.) DISPLAY-MODIFY-PARMS NVE-lO NVEIBIO DTNOR-BETA*DX1**2/AMAX1(DRED:DOX) DT=TRISEI200. IF (DTNOR.LT.DT) DTBDTNOR TIME-0.0 TOUT-DTOUT DO (1'1:100) . C(I)-COX CI(I)=COX P(I)-CRED PI(I)-CRED DX(I)=DX1*EXP(GGO*FLOAT(I)) .FIN EEOUILI-25.691*ALOG(CRED/COX) EBEEQUIL+ESTEP KFIBKSTD*EXP(-ALPHA*E/25.691) KBI-KSTD*EXP((1.-ALPHA)*E/25.691) WRITE (7:100)_”38:”110:'33:”112 000 000 176 FLECS/RT-ll V28.01 PAGE 3 NONIDEAL SIMULATION ROUTINE REPEAT UNTIL (TOUT.GT.TFINAL) . IF (TIME.GT.2.*TRISE) DT=DTNOR WHEN (TIME.GE.TRISE) E-EEGUIL+ESTEP ELSE E-EEGUIL+ESTEP*(TIME/TRIBE) KF‘KSTD*EXP(-ALPHA*E/25.691) KB-KSTDfiEXP((1.~ALPHA)*E/25.691) SURFACE BOUNDARY CONDITIONS FLUX 3 (KF*C(1)-KB*P(1)) / I (1.+KF*DX1/DOX+KB*DX1/DRED) C0-C(1)-FLUX*DX1/DOX PO=P(1)+FLUX*DX1/DRED DIFFUSION CNEH(l)-C(1)+DOX*DT/((DX(2)+DX1)*.5) * ((C(2)-C(1))/DX(2)-(C(1)-CO)/DX1) DO (182:NVE) CNEN(I)-C(I)+DOX*DT/((DX(I+1)+DX(I))*.5) * . ((C(I+1)-C(I))IDX(I+1)-(C(I)-C(I-1))IDX(I)) ...FIN DO (I-l:NVE) C(I)-CNEH(I) CNEN(1)=P(1)+DRED*DT/((DX(2)+DX1)*.5) * ((P(2)-P(1))lDX(2)-(P(l)-PO)/DX1) DO (182:NVE) . CNEU(I).P(I)+DRED*DT/((DX(I+1)+DX(I))*.5) * ((P(I+1)-P(I))IDX(I+1)-(P(I)-P(I-1))IDX(I)) ...FIN DD (131:NVE) P(I)-CNEN(I) IF (ABS(C(NVE-2)-COX).GT.COX*.001) NVEBNVE+1 IF (ABS(P(NVE-2)-CRED).GT.CRED*.001) NVEBNVE+1 IF (NVE.GT.100) STOP ’TOO MANY VOLUME ELEMENTS’ 000 000 “I" 177 FLECS/RT-ll V28.0l PAGE 4 IDEAL SIMULATION ROUTINE UNLESS (TIME.EG.0.0) KF-KFI . KB-KBI ...FIN SURFACE BOUNDARY CONDITIONS FLUXI I (KF*CI(1)-KB*PI(1)) / (l.+KF*DX1/DOX+KB*DX1IDRED) COI-CI(1)-FLUXI*DX1IDOX POI!PI(1)+FLUXI*DX1/DRED DIFFUSION CNEN(1)-CI(1)+DOX*DTI((DX(2)+DX1)*.5) i ((CI(2)-CI(1))lDX(2)-(CI(1)-COI)/DX1) DO (1.2:NVEI) CNEN(I)-CI(I)+DOX*DT/((DX(I+1)+DX(I))fl.5) * ((CI(I+1)-CI(I))/DX(I+1) - . (CI(I)-CI(I-1))/DX(I)) ...FIN DO (1-1:NVEI) CI(I)'CNEN(I) CNEH(1)-PI(1)+DRED*DT/((DX(2)+DX1)*.5) * ((PI(2)-PI(1))lDX(2)-(PI(1)-POI)/DX1) DO (182:NVEI) . CNEH(I)-PI(I)+DRED*DT/((DX(I+1)+DX(I))*.5) * ((PI(I+1)-PI(I))/DX(I+1) - (PI(I)-PI(I-1))/DX(I)) ...FIN DO (1'1:NVEI) PI(I)=CNEH(I) IF (ABS(CI(NVEI-2)-COX).GT.COX*.001) . NVEI-NVEI+1 ...FIN IF (ADS(PI(NVEI-2)-CRED).GT.CRED*.001) . NVEIINVEI+1 ...FIN IF (NVEI.GT.100) . STOP ’TOO MANY VOLUME ELEMENTS (I)’ .FIN 000 178 FLECS/RT-11 V28.01 OUTPUT SECTION IF (TIME+.5*DT.GT.TOUT) TOUT-TOUT+DTOUT CURRIFLUX¥964S7.E6*AREA RATBFLUX/FLUXI TYPEioTIMEoCURRoRAT:NVE .FIN NEXT TIME INCREMENT . TIME-TIME+DT ...FIN . PAUSE ...FIN STOP PAGE 5 110 100 l-‘H huhpuuuwuuupuv— 179 FLECS/RT-11 V28. 01 PAGE 6 USER I NTERF ACE SECT I ON TO DISPLAY-MODIFY-PARMS WRITE-DATA REPEAT UNTIL (I.EG.O) WRITE (7:110) ”33:”131:“61:”40:'83:“112 FORMAT (1X:6A1:’ENTER ENTRY TO CHANGE:’$) ACCEPT*:I CONDITIONAL (I.LT.O) WRITE-DATA (I.GT.14) “RITE-DATA (I.E0.0) CONTINUE (OTHERWISE) TYPE*:’ENTER NEH VALUE’ ACCEPT*:A(I) IIIII+"41 “RITE (7:120) “33:”131:III:“64:A(I) ...FIN ...FIN ...FIN .FIN TO "RITE-DATA WRITE (7: 100) “33: "110: '83: “112: KSTD: ESTEP: COX: DOX: CRED: DRED:: ALPHA AREA: DX1: GOG: BETA: DTOUT: TFINAL: TRISE FORMAT (1X:4A1: ’GUASI-REV CHRONOAMPS’: // 1X: ’ 1 STD RATE CON’: T22: 1PO10. 3: 1X: ’ 2 E STEP’: T22: 010. 3/ 1X: ’ 3 CONC OX’: T22: 010. 3/ 1X: ’ 4 DIFF COEFF OX’: T22: 010. 3/ 1X: ’ 5 CONC RED’: T22: 010. 3/ 1X: ’ 6 DIFF COEFF RED’: T22: 010. 8/ 1X: ’ 7 ALPHA’: T22: G10. 3/ 1X: ’ 8 AREA’: T22: 010. 3/ 1X: ’ ? 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