THEQRY ARE APPLICATEGN C}? THE F‘G'FENTEAL STEP-LENEAR KAN ELECTRC-LYSES’ MSW-ESE; {ééi‘éETECS ARE? MEfiHANE$M Qfi EEQUCWQK GE: 338§?§?UTED AZQEfiENZENE QGMFQUNBS Hints {:09 Has Duqm 0.5 p5. D. MICHIGAN SMTE UNIVERSITY Joseph Theodore Lmdquist, Jr. 1967 [Ht-.513 This is to certify that the thesis entitled THEORY AND APPLICATION OF THE POTENTIAL STEP-LINEAR SCAN ELECTROLYSIS METHOD. , KINETICS AND MECHANISM OF REDUCTION OF SUBSTITUWS‘AE‘Qfilm-IZENE COMPOUNDS Joseph Theodore Lundquist, Jr. has been accepted towards fulfillment of the requirements for Ph.D. Chemistry _— degree in _____' I/T/ / 1" x/ k» 57* 1 Major professor Date November 27, 1967 0-169 3. 18 R A 3?: Y ltéichigm State 3,,h University t .m— e— 1, WW1- //:/.g, (/73: , / <2 ._ "W: mg... .q..-. -.—4— ., ”*- ABSTRACT THEORY AND APPLICATION OF THE POTENTIAL STEP-LINEAR SCAN ELECTROLYSIS METHOD. KINETICS AND MECHANISM OF REDUCTION OF SUBSTITUTED AZOBENZENE COMPOUNDS by Joseph Theodore Lundquist, Jr. Theory for the potential step—linear scan technique for the case of a chemical reaction following reversible electron transfer has been developed without the use of simplifying approximations introduced by previous authors. Results of the theoretical calculations are presented in the form of working curves from which homogeneous kinetic parameters are easily determined. Results of the theory indicate that this technique should be a versatile method for studying electrode processes and measuring homogeneous rate constants. These conclusions are illustrated by mea- suring rates of benzidine rearrangement of hydraZObenzene and mfhydrazotoluene. In the case of the latter compound a direct comparison is given between rate constants measured electrochemically and by classical kinetic methods, and it is concluded that homogeneous rate constants can be measured electrochemically with complete confidence. The mechanism of the electrode reaction for reduction of azobenzene also has been investigated in detail in aqueous and non-aqueous solvents. Because of the unambiguous mech- anism found in N,N-dimethylformamide several substituted azobenzene compounds were studied in an attempt to correlate Joseph Theodore Lundquist, Jr. reactivity with structure. Both half—wave potentials and standard heterogeneous rate constants gave linear correla- tions with Hammett g values. To explain results obtained in aqueous solutions a general mechanism was proposed where hydrogen ions partici- pate in preceding and succeeding chemical equilibria and also in the rate determining step of the electrode reaction. Theory of cyclic voltammetry was extended to include this general mechanism. Experimental results for reduction of azobenzene were found to be ccnsistent with two electrons and two hydrogen ions involved in the rate determining step. However, additional experimental data that suggest stepwise reduction, which also would be consistent with the cyclic voltammetric data, are presented. THEORY AND APPLICATION OF THE POTENTIAL STEP-LINEAR SCAN ELECTROLYSIS METHOD. KINETICS AND MECHANISM OF REDUCTION OF SUBSTITUTED AZOBENZENE COMPOUNDS BY Joseph Theodore Lundquist, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1967 . ,.- o 1’ (1' f) ‘1 W W D I a J t \J g ' VITA NAME: Joseph Theodore Lundquist, Jr. BORN: May 2, 1940 in Bay City, Michigan ACADEMIC CAREER: Midland High School Midland, Michigan (1954-1958) Michigan State University East Lansing, Michigan (1958-1967) DEGREES HELD: B.S. Michigan State University (1963) M.S. Michigan State University (1965) ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. Richard S. Nicholson for his guidance and encouragement throughout this study. The author is grateful to the Socony Mobil Oil Company, United States Army Research Office—-Durham under Contract No. DA—31-124—ARO—D-308, and the Department of Chemistry for financial aid. Special thanks go to the author‘s wife, Joann, for assistance rendered throughout the course of his graduate studies. ii TABLE OF CONTENTS Page INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1 DESCRIPTION OF POTENTIAL STEP—LINEAR SCAN TECHNIQUE. 3 THEORY . . . . . . . . . . . . . . . . . . . . . . . 9 Solution of the Integral Equation . . . . . . 14 Results of Numerical Calculations . . . . . . 17 Measurement of Rate Constants . . . . . . 26 EXPERIMENTAL EVALUATION OF POTENTIAL STEP-LINEAR SCAN THEORY . . . . . . . . . . . . . . . . . . . 30 EXPERIMENTAL . . . . . . . . . . . . . . . . . . . . 33 Instrumentation . . . . . . . . . . . . . . . 33 Chemicals . . . . . . . . . . . . . . . . 43 Procedures . . . . . . . . . . . . . . . . 43 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . 45 Effects of Gelatin . . . . . . . . . . . . . . 45 Reduction of Azobenzene . . . . . . . . . . . 58 Reduction of mfAzotoluene . . . . . . . . . . 65 Comparison of Rate Constants . . . . . . . 69 REDUCTION OF AZOBENZENE IN N1N-DIMETHYLFORMAMIDE . . 71 EXPERIMENTAL . . . . . . . . . . . . . . . . . . . . 74 Materials . . . . . . . . . . . . . . . . . . 74 Solvent . . . . . . . . . . . . . . . . . 74 Chemicals . . . . . . . . . . . . . . . . 74 Procedures . . . . . . . . . . . . . . . . . . 75 Solutions . . . . . . . . . . . . . . . . 75 Evaluation of Diffusion Coefficients . . . 77 Evaluation of El . . . . . . . . . . . . 77 Measurement of Heferogeneous Rate Constants 77 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . 80 Evaluation of Diffusion Coefficients . . . . . 84 Comparison of Measured Diffusion Coef- ficients with Stokes-Einstein Diffusion Theory . . . . . . . . . . . . . . . . . 84 Effect of Double Layer Structure on Heterogeneous Rate Constants . . . . . . . . 88 iii TABLE OF CONTENTS (Cont.) Page Applicability of the Hammett Equation . . . 90 Variation of Half—Wave Potentials with Structure . . . . . . . . . . . . . . . . . 94 Variation of Heterogeneous Rate Constants with Structure . . . . . . . . . . . . . . 95 MECHANISM OF REDUCTION OF AZOBENZENE IN WATER . . 100 THEORY . . . . . . . . . . . . . . . . . . . . . . 101 EXPERIMENTAL . . . . . . . . . . . . . . . . . . . 105 RESULTS AND DISCUSSION . . . . . . . . . . . . . . 106 LITERATURE CITED . . . . . . . . . . . . . . . . . 122 APPENDIX A . . . . . . . . . . . . . . . . . . . . 126 APPENDIX B . . . . . . . . . . . . . . . . . . . . 128 iv LIST OF TABLES TABLE Page I. Variation oftJT Xp with k/§_and 2A, . . . . . 28 II. Polarographic data for the reduction of azobenzene at various gelatin and acid concentrations . . . . . . . . . . . . . . . 47 III. Comparison of rate constants for rearrangement of hydrazobenzene determined with and without gelatin present . . . . . . . . . . . . . . 51 IV. Diffusion coefficients of azobenzene for different acid and gelatin concentrations . 55 V. Rate constants for rearrangement of m: hydrazotoluene . . . . . . . . . . . . . . . 66 VI. Comparison of observed and literature values for melting points of substituted azobenzene compounds . . . . . . . . . . . . . . . . . 76 VII. Diffusion coefficients of substituted azo- benzene compounds in N,N-dimethylformamide . 84 VIII. £4 and Es values for azobenzene compounds i 2N,N-dimethylformamide . . . . . . . . . . 92 IX. Variation of EJ/z and (kg) with hydrogen app ion concentration for reduction of azobenzene in aqueous solutions . . . . . . . . . . . . 107 Figure 1. 10. 11. 12. 13. 14. LIST OF FIGURES (a) Applied potential as a function of time for potential step—linear scan method; (b) current—time curves for large and small rate constants . . . . . . . . . . . . . . . . . . Variation of current function with k/§_(0.1, 0.05, 0.01) for §A_= 15.. . . . . . . . . . . Variation of anodic peak potential with k/§_ for 1A = 0.5 and 62.5 . . . . . . . . . . . . Variation of peak current function with k/a for 2A.: 0.5, 2.5, 7.5, 20.0, 37.5 and 62.5 Circuit diagram of potentiostat with floating load resistor . . . . . . . . . . . . . . . . Circuit diagram of modified potentiostat . . Cell arrangement . . . . . . . . . . . . . Conventional polarographic current-potential curves showing effects of gelatin on reduction of azobenzene . . . . . . . . . . . . . . . . Stationary electrode polarograms showing effects of gelatin on reduction of mM azo- benzene (scan rate, 30 mv sec 1) . . . . . . Chronoamperometric curves showing effects of gelatin on reduction of azobenzene . . . . . Comparison of theory and experiment for poten- tial step—linear scan method . . . . . . . . Variation of k_with Ho for the benzidine rearrangement of hydrazobenzene . . . . . . . Variation of k with concentration of per- chloric acid for rearrangement of m-hydrazo- toluene. . . . . . . . . . . . . . . . . . . Cyclic stationary electrode polarogram for reduction of mM azobenzene in 0.3M tetra— ethylammonium— perchlorate and N,N-dimethy1- formamide (scan rate, 40 v sec‘l) . . . . . vi Page 18 21 24 34 37 40 49 52 56 6O 63 67 81 LIST OF FIGURES (Cont.) Figure 15. 16. 17. 18. 19. 20. 21. 22. Variation of diffusion coefficient of substituted azobenzene compounds with _ molecular weight (error level, 10.2 x 10 6 cm2 sec‘l) . . . . . . . . . . . . . . . . . Variation of g; with Hammett g_value for substituted azo enzene compounds (slope, 0.36, volts; gJ/z y§_S.C.E.) . . . . . . . . . . . Variation of k with Hammett g value for sub- stituted azobenzene compounds (slope, 0.43 i 0.03; error level, i 10%) . . . . . . . . . . Variation of g; with perchloric acid con- centration for ééduction of azobenzene (slope, 56 mv per unit pH change; EJ/z XE S.C.E.) . . Variation of (ks)app with perchloric acid concentration for reduction of azobenzene (Slope, 0.76) O O O O O O O O O O O I O O O 0 Variation of (ks)app with [H+]°-76 for re- duction of azobenzene (slope, 0.105 cm sec-1) Comparison of theory and experiment . . . . . Cyclic stationary electrode polarograms for reduction of 0.4 mM_azobenzene in 0.3M_tetra- ethylammonium perchlorate and N,N—dimethyl- formamide with varying amounts of benzoic acid (scan rate, 40 v sec-1) . . . . . . . . vii Page 86 93 97 108 110 112 115 118 INTRODUCTION Whenever a chemical reaction can be initiated electro- lytically > R 0 + ne < R > Z electrochemical techniques possess some unique advantages over more classical approaches for studying kinetics of the chemical reaction. For example, because the reacting inter- mediate, R, is generated at the electrode, this same elec- trode can be used to follow the concentration of R, and as a result kinetics of very rapid chemical reactions can be measured. Although several electrochemical techniques have been developed for studies of this type, many of these tech- niques are limited for one reason or another. One method which appeared to have a minimum of limitations is the potential step—linear scan technique. Although this method seemed to be potentially very important, a quantitative development had not been reported, and therefore this task was adopted as the major goal of this thesis. Thus, the majority of the thesis is devoted to theoretical development and experimental applications of the potential step-linear scan method. The experimental applications involve studies of ben— zidine rearrangements which are initiated by reduction of azobenzene. For purposes of evaluating the potential step— linear scan method these reductions were performed under 1 2 conditions where the electrode reaction is reversible. Be- cause of this no information about the actual mechanism of the electrode process was required, or obtainable for that matter. However, after establishing the usefulness of the potential step-linear-scan technique, it was logical to attempt heterogeneous kinetic studies of the electrode re- action in an effort to determine its mechanism. To do this it also was necessary to determine the role of hydrogen ion, and therefore experiments were performed in an aprotic sol- vent as well as in water. Results of these mechanistic studies also are included in this thesis. DESCRIPTION OF POTENTIAL STEP-LINEAR SCAN TECHNIQUE Several different electrochemical techniques are suit- able for investigation of Mechanism I. These methods differ primarily in the form of the external perturbation applied to the electrode, and can be classified as either one-step or two-step methods. The one-step techniques are exempli- fied by conventional polarography (1—4) where removal of R from the electrochemical equilibrium results in a shift of El/z and distortion of the current—voltage curve from that expected for reversible uncomplicated reduction. The magni- tude of these effects is related to the rate constant, 5, These indirect methods are of limited use, however, because the above parameters are relatively insensitive to changes of k, and also El/z in the absence of kinetic complications must be known. With the two—step methods R is generated at a controlled rate during the first step, and then unreacted R is measured in the second step. The change in the amount of R observed relative to the case with no kinetic complications gives a measure of 5, These two-step techniques have the advantage that it is unnecessary to know EJ/z in the absence of kinetic complications. Typical of these methods are chronopotentio- metry with current reversal (5-8), controlled potential step-functional electrolysis (9), and cyclic voltammetry (10). A fourth technique, the potential step-linear scan method, is a combination of the last two methods—-i.e., a voltage 3 4 step is applied to the electrode, followed by a linear potential scan (Figure 1a). To explain qualitatively the effect of Mechanism I on a potential step-linear scan experiment, it is useful to consider the two parts of the experiment separately. The first part (t_j,AJ see Figure 1) consists of a controlled potential generation of the intermediate, R. The second part (£_> A) is the measurement step in which stationary electrode polarography is used to analyze for unreacted R. During the generation step the current will follow normal diffusion controlled decay, provided potential is stepped to a value corresponding to the limiting current region (Figure 1b). Current during this part of the experiment is unaffected by the chemical reaction, because at limiting cur- rent potentials, current is determined only by the rate of arrival of substance 0 at the electrode surface. However, during the measurement step currents should be markedly af- fected by the chemical reaction. Consider, for example the case where the chemical reaction is slow (or, equivalently, scan rate is relatively large). The current for this case should be large, because essentially all of R remains in the vicinity of the electrode (see Figure 1b). As the rate of the chemical reaction increases, however, the resulting current should decrease because of the decrease of R near the electrode. By comparing this decrease of current with the diffusion controlled current (t_< A), it should be pos- sible to determine the rate constant, k. Figure 1. (a) Applied potential as a function of time for potential step—linear scan method; (b) current— time curves for large and small rate constants. Hmflucmuom > Time large k Time small k pamHHdU Figure 1 7 Based on the above discussion apparently the potential step-linear scan technique should possess some unique ad- vantages over other electrochemical methods. For example, some of the best features of both square-wave electrolysis and cyclic voltammetry are combined. The method has the important advantage of square-wave electrolysis that ad- sorption effects are minimized because reduction of adsorbed material occurs only during the initial fraction of the experiment (9). The method has the additional advantage of cyclic voltammetry that the potential dependence of oxida- tion of R is retained as an experimental parameter. This potential dependence provides important information about the activated product, R, similar to polarographic half- wave potential data, but on a much broader time scale (10). Because of the unique features of the potential step- linear scan technique, and its potentially important applica- tion in electrochemical studies, it seemed important to attempt a rigorous theoretical development of the method. During the course of this work, a theoretical treatment was presented by Schwarz and Shain (11). The theory of these authors, however, is inexact, and because of simplifying approximations, application of their results is limited to cases in which duration of the linear scan is small with re- spect to the half-life of the chemical reaction (11). This means that scan rates must be used which are as much as two orders of magnitude larger than actually necessary to detect kinetic effects. Consequently, factors such as charging 8 current and charge transfer kinetics rapidly become limiting. In addition, it always is necessary to measure small changes of large peak currents, and considerable sensitivity is lost. For these reasons the theory of Schwarz and Shain is of limited usefulness compared with the rigorous treatment pre- sented here. THEORY Development of a mathematical description of the poten- tial step—linear scan method for Mechanism I is equivalent to calculating the concentration profiles of substances 0 and R. These concentration profiles depend on the potential perturbation employed (potential step-linear scan), and the mass transport process. Of the various types of mass trans— port, diffusion is the simplest to describe mathematically. Fortunately, by employing conventional polarographic condi— tions, it is possible to limit experimentally the mass transport process to diffusion, and therefore Fick's dif- fusion equations can be used to calculate concentrations of 0 and R (12,13). The diffusion equation for R, of course, ,must be modified to account for homogeneous kinetic reaction of R (10). Application to substances 0 and R of Fick's second law for linear diffusion gives BC (x,t) 52C (x,t) —-g————° = D 0 (1) t 0 8x2 ac (x,t) 82c (x,t) R _ 4R _ “‘5?“ *DR BX, “129“) , (2) where CO(x,t) and C x,t) are concentrations of O and R( R respectively, $.15 distance from the electrode surface, t'is time, DO and DR are diffusion coefficients, and k_is .the first order rate constant for the homogeneous chemical reaction. ' 9 10 Solutions of Equations 1 and 2 contain integration constants which must be evaluated. To determine these con- stants initial and boundary conditions, which are mathema— tical statements of experimental conditions, must be formu- lated. Initially concentrations of 0 and R are assumed to be equal to their bulk analytical values t=O7 X_>_O: C X.C) = C* : CR(x,o) = C§(r~JO) (3) o( 0 After the experiment is started it is assumed that a point sufficiently removed from the electrode surface exists beyond which there are no concentration gradients. This condition is readily satisfied, even for cells of small dimension (12). Mathematically this condition is t > 0 ; X -—> GD: * *- C0(x,t) ——> CO ; CR(x,t) ——» cR (4) A material balance equating rates of arrival of O and departure of R from the electrode surface provides the third boundary condition t > 0 ; X = 0: ac (x,t) BC (x,t) D 0 = -D R (5) 0 5x R 5x To obtain the final boundary condition it will be as- sumed that the redox couple obeys the Nernst equation. To assume a more general potential-concentration relationship 11 would complicate the theory considerably (10,14). Moreover, it usually is possible to adjust conditions experimentally so that this assumption is satisfied. Thus, applying the Nernst equation to the redox couple, one obtains C 0,, = as [(3%) (E - 1:20)] (6) where .3 is the number of electrons, E. is electrode poten- tial, E9 is the formal electrode potential, and R, T and F have their usual meaning. For the potential step-linear scam technique Equaticn 6 is rewritten to account for variation of electrode potential with time (Figure 1a). The result of these operations is Equation 7 (nF/RT)(Ei - E0) t = 0 (7a) ln(S%(at)) = (nF/RT)(Ei - E0 - ES) 0 < t.: 1 (7b) (nF/RT)(Ei - EO - ES) + at - ah t > A (7c) where Sx(at) = CO(C't)/CR(O,t) (7d) There E1 is initial potential, E5 is the potential to which the electrode is stepped, .A is time at which linear scan begins (Figure 1a), and .2 is a parameter directly proportional to the scan rate, y! during the linear scan. _ an 12 To obtain general solutions of the above boundary value problem, it is useful first to reduce it to integral equation form (10). This reduction can be accomplished with the aid of the Laplace transformation (10). Hence Laplace trans- formation of Equations 1 to 3 yields 52C°‘(X'S) < > < > D .7 = s c x,s — c* 9 0 5X2 0 O 52CR(X.S) D = (s - k) c (x,s) (10) R a 2 R x where C(xjg) is the Laplace transform of .C(§Jt). Equations 9 and 10 are ordinary differential equations, which,together with Equation 4, can be solved to give *- Co 0 *- .12 S C0(x,s) (co(o,s) — ) exp [-X(%b)1/2] + (11) S+k cR(x.s> cR(os> exp [-x ( W21 (12) R By differentiating Equations 1 and 12 with respect to x. and then evaluating at x_= 0, one obtains ac0(x,S) cg DO T Xzo = - (CO(O,S) - §—) \(DOS (13) 5CR(x,S) DR T X=O - - CR(o,s) JDR(s + 1.7 (14) The left hand side of Equations 13 and 14 is the Laplace and R transform of the surface flux of O Inversion of Equations 13 and 14 reSpectively. is accomplished with the convolution theorem of operational calculus (15). 13 Inversion followed by application of Equation 5 yields the following integral relationships between surface concentra- tion and flux _ T dT cO(o,t) - (15) o t - T C (O t) = _._I_ fU) eXP{“kLt - T)] (11 (16) R I rs/erO o JET? where D O V (17) DR and BC (x,t) . Equation 18 is a statement of Fick's first law where A_ is electrode area. Substitution of C (0,t) and C 0,t) (Equations 15 O and 16) into Equation 7d gives R( f§T2d1_yS7\(at) ft HQ eXp{-k(t - «()de (19) J's—.7 37:5,; 0 fr..— Equation 19 is a linear Volterra integral equation with variable coefficients (16), and represents a succinct state— ment of the entire boundary value problem. ~Further opera- tions on Equation 19 are facilitated if it is made dimension- less by the following changes of variable (10) T = z/a (20) and 14 f(t) = CSIJwaDO x(at) (21) Equation 19 now takes the dimensionless form, at _f:__ xgzgdz _ vs“. )f X(2) eXp(- <1;- -—>.A the curves are only slightly dependent on E but even in __s I this case ES effectively can be eliminated as a variable in Equation 22 (see discussion under Measurement of Rate 18 Figure 2. Variation of current function with 25/31 (0.1, 0.05, 0.01) for 2.1 = 15. 19 -O.L_. Aumvx $5 -0.3— 28 2O 12 at Figure 2 20 Constants). All calculations reported here were performed with the arbitrary value of (El/2 - Es)n_ equal 167 mv. Effect of k/a and ah. In general both the parameters k/aland'aA influence solutions of Equation 22. For example, Figure 2 illustrates the effect of the kinetic parameter k/§_for a fixed value of 3A, Clearly for times less than '5 the current is independent of 313- This fact is consistent with Equation 30, and is a result of the stepping potential used. As k a_ approaches zero the linear scan portion of the curve also becomes independent of k/a_ provided measure— ments are made to the extension of the current—time curve as a baseline. For measurements made in this way the linear scan portion of the curves also is independent of .3A in a manner analogous with cyclic voltammetry (10). For example, for small k/a_ the value of "JFXp* always approaches 0.446, which is the same value obtained for cyclic voltammetry in the absence of kinetic complications (10). For finite values of k/a_ the linear scan portion of the curve is sensitive to changes of 5/3 in two ways. First, the peak potential shifts anodically as k/§_ increases. This behavior is illustrated in Figure 3 where (gp — El/2)n is plotted as a function of k g_ for two values of .al. Although peak potential clearly depends on both k/a and 2A, *, The new subscript refers to the peak value of the function JTx(at) measured to the extension of the current-time curve. 21 Figure 3. Variation of anodic peak potential with 5/3 for a); = 0.5 and 62.5. 22 30.— 40—— 70)— 80-— 62. —1.5 —1.0 log (k/a) Figure 3 23 the dependence on 33 is not great since the values of 3A, represented in Figure 3 correspond to extreme limits. The second effect of 3/3 is in terms of anodic peak current, and is illustrated in Figures 2 and 4. In Figure 4 JTXP is plotted versus 3/3 for several values of 3A. In general for fixed 3hy~fTXp decreases with increasing 3(3. The actual magnitude of the peak, however, is a function of both k/3_and 33. Thus, for fixed 3/3 the size of the anodic peak also decreases with increasing 33. The reason for this effect is that with large 33 (constant potential step cor- responding to long times) and finite 3/3, more time is avail- able for the chemical reaction to proceed, and therefore the concentration of unreacted R available for oxidation during the linear scan is less. From an experimental point of view the fact that.JTxp is a function of both 33 and 3/3 is particularly important, because this allows adjustment of two experimental parameters, A and y] in such a way as to optimize influence of the kinetics. Thus, to study very fast chemical reactions does not necessarily require large scan rates, because 33 can be made small. In this case (see Figure 4, small 33) kinetic effects are measurable for relatively large 3/3 (i.e. slaw scan rates). This situation is in sharp contrast to the work of Schwarz and Shain (11) where their theoretical re— sults are restricted to working with values of 3/3 less than about 0.008. Therefore, to measure the same value of‘k using the theory of Schwarz and Shain (11) requires scan 24 ll. I.I. .m.mo pom .m.bm .o.om .m.h .m.m .m.o n Km How M\M LDHB Goauocsm DCTHHSU Mmmm mo coaumaum> .v mudmflm 25 v ousmflm ¢w\xv moH m.ml o.N| m.Hn o.HI m.ou q _ _ _ _ 26 rates approximately 2 orders of magnitude larger than is required using data such as those of Figure 4. An analo- gous situation exists for measuring small chemical rate constants, where large values of 33 (see Figure 4) can be selected to provide enhancement of kinetic effects. Measurement of Rate Constants. To measure rate con- stants quantitatively, data such as those of either Figure 3 or 4 could be used. From an experimental point of View peak potential shifts are relatively small, and therefore it is preferable to apply the data of Figure 3 as a diag— nostic criterion to ensure that the system under considera— tion conforms with Mechanism I. As implied in the preceding section, data such as those of Figure 4 are most suitable for quantitative measurement of 5“ but to do this experimental values of *JTXP must be obtained. This is easily accomplished by recognizing that i p “\ m where 2p is the experimental peak current (measured to the extension of the current-time curve), 33 is the current at /\ time by and the remaining terms already have been defined. Thus, data of Figure 4 serve as working curves from which experimental values Of lp/l: (1/‘F37) can be converted I'\ I . \ directly to values of 5fl3: since 3_ is known, 3. can be calculated. In addition to its simplicity this approach has the advantage that experimental parameters such as C8, D0' 27 A, etc., need not be known. Table I contains numerical values from which working curves like those of Figure 4 can be constructed. It should be emphasized that the curves of Figure 4 (and data of Table I) are strictly applicable only when (El/2 - ES)3_ equals 167 mv. Actually the linear scan por- tion of the curves is not strongly dependent on E6, and values of (31/2 — Es)g_within about 110 mv of 167 mv are acceptable. Nevertheless, this fact does constitute a limitation of the method, and therefore an alternate ap- proach was sought. This approach consists of realizing that if a stepping potential more cathodic than —167 mv is used, then an effectiVe A, Ae’ can be defined as the time at which the electrode potential during the anodic Span has reached a value of -167 mv with reapect to 31/2. This is possible because of the fact that for any potential more cathodic of EJ/z than —167 mv, the electrode potential always is in the limiting current region and the curve is described by Equa- tion 30 regardless of whether the potential is fixed or is a function of time (22). Because peak potential of the anodic curve is directly proportional to EJ/z for 3/3 < 0.1 (see Figure 3), in this case fie is defined as the point 197/3 mv cathodic of the anodic peak. If values of 3/3_greater than 0.1 are encountered the correct estimate of Ae can be ob— tained from Figure 3. Experimentally, then, one simply en— sures that (31/2 - E )3 is greater than or equal 167 mv, and “as after the experiment is completed the value of le to be used with Figure 4 is determined. 28 woo.o ooov.o mwooo ooow.o oHHoo ooowoo wmooo oomfioo hoaoo ooomoo wmfioo commoo mwfioo ooowoo mhfioo OQOQOQ meoo 00mhoc Hmfloo oomo.o ovfioo oooaoo ooaoo oo¢H.o whaoo Qummoo mwaoo ooowoo an.o ooom.o mmfloo oomcoo mmfioo oomo.o mHm.o ommo.o Hmaoo ooom.o mmaoo commoo Homoo ooow.o mmm.o ommooo mmfioo mmmo.o omm.o oomo.o wmm.o ommfioo ommoo ooomoo ammoo oomm.0 mwm.o omHo.o mmN.o oowo.o mamoo oomo.o wmm.o ooofloo mwm.o oomfioo mmmoo ooom.o «mmoo OOHOOO Humoo Nmmooo Nwmoe OONQOO ammoo ommooo nmm.o Oowmoo meoo oomfioo mmmoo nmoooo wmm.o owfiooo mmmoo OOHQOQ mwmoo oewooo mvmoo .onooo mmmoo OOOHoo mmm.o omoo.o mmm.o ooaooo mHv.o omoo.o mwmoo mmm0.0 wwm.o oowo.o wwmoo oomo.o mawoo «moooo «mv.o hmoo.o wmwoo omoooo movoo mmHo.o QHwoo oomooo mmm.o oovooo omwoo oaoooo owvoo oaoooo mvvoo oaoooo mmvoo mmoooo mmvoo ooaooo mav.o ommo.c fiXkfi m\x mX$5 M\x mxts. M\M mx¢3 m\x mxfis. M\M mekl m\x m.mm n.5m om m.h m.m m.o .Ll Km .fim.bcm.M\M QHHB mxts.mo coaumanm> .H magma 29 Results of calculations described above extend the versatility of the potential stepmlinear scan method con- siderably with respect to the theory of Schwarz and Shain (11). Because application of the rigorous theory is easier than application of the original theory of Schwarz and Shain (11), the potential step-linear scan method now pro— vides a very attractive means of measuring rate constants of chemical reactions initiated electrochemically. EXPERIMENTAL EVALUATION OF POTENTIAL STEP~LINEAR SCAN THEORY To test experimentally the theory of the potential step—linear scan method develOped above requires a compound that is reduced (or ox1dized) according to Mechanism 1. and for which all of the kinetic parameters have been evalu- ated by an independent method. One compound which satis— fies these criteria is azobenzene. A number of previous electrochemical studies (23-33) have established that azo- benzene is reduced reversibly in acidic solutions to hydrazo- benzene which undergoes the benzidine rearrangement (34). The benzidine rearrangement is an intramolecular, ir— reversible reaction, and in the case of azobenzene the rearranged products, benzidine and diphenyline, are not electroactive. Moreover, kinetics of the benzidine rear— rangement of hydrazobenzene have been evaluated with several different electrochemical techniques (9 11,35). Thus, azo- benzene appeared to be ideally suited to test the theoretical calculations, and evaluate the pOtential step-linear scan technique. In addition to testing the theory deveIOped for the potential step-linear scan method, these studies of benzidine rearrangements provided the Opportunity to compare homo- geneous rate constants measured electrochemically with ones determined by more cla881cal approaches. In spite of the number of cases in which homogeneous rate constants have 30 31 been measured electrochemically, there essentially are no examples in the literature where a direct comparison has been made with rate constants measured by accepted classi- cal methods. The major reason for this unusual situation is that the time scales of the two approaches generally do not overlap. Electrochemical techniques are useful for measurements on a much shorter time scale than classical approaches. Thus, provided the electrochemical techniques give meaningful rate constants, the two approaches are complementary. Because of these differences of time scales, comparisons that have been attempted involved an extrapola- tion of one of the sets of data. For example Schwarz and Shain (9) used the Hammett acidity function to extrapolate electrochemically measured rate constants about two orders of magnitude (toward longer half-lives) to compare them with classical measurements. More recently, Reilley and co- workers (35) applied dielectric constant corrections to classically measured rate constants to provide a comparison with an electrochemically determined constant. Results in both of these cases indicate reasonable agreement between the two approaches. Nevertheless, the uncertainties as- sociated with the various extrapolations leave room for argument that the agreement may have been in part fortuitous. For these reasons the potential step-linear scan theory also was used to measure rate constants for rearrangement of mfhydrazotoluene. This compound was selected because existing literature data (36) for rate constants measured 32 spectrophotometrically appeared to provide the best op- portuntiy for comparison of rate constants. An extrapola- tion of some of the classical data based on dielectric cor- rections was used, but in addition some of the spectro- photometric experiments were repeated to prove that the extrapolations are valid. The electrochemical reduction of azobenzene and mfazo— toluene on a mercury electrode is complicated by adsorption (11,37). Although the potential step-linear scan technique has the advantage that adsorption effects are reduced, for generation times much greater than the half-life of the chemical reaction, anomalous surface phenomena were observed. The addition of gelatin to solutions of these compounds reduced these effects so that longer generation times could be used. To justify using gelatin a study of its effect on the reduction of azobenzene also was made. EXPERIMENTAL Instrumentation Ohmic potential losses are a possible source of seri— ous error in all of the electrochemical measurements made in this study. Generally, correction for these ohmic potential losses cannot be accomplished by simply applying Ohm's law (38). However, by using a three electrode poten- tiostatic circuit it is possible to compensate electronically for a majority of the potential loss. Figure 5 is a block diagram of the potentiostatic circuit which was used to accomplish this compensation. I The control amplifier (C.A.) is a high gain differential amplifier which employs negative feedback to maintain the difference of potential between the inverting (-) and non- inverting (+) inputs at zero volts. Since the sum of the potentials in the comparison loop (inverting input, refer- ence electrode (R.), working electrode (W.), function genera- tor (F.G.) and noninverting input) is zero, any potential of the function generator will be imposed between the refer- ence and working electrodes. Two different control amplifiers were employed, a com- mercial instrument (Wenking Potentiostat, Model 61RS, Brink— mann Instruments, Westbury, N.Y.), and one constructed from operational amplifiers and a booster (Philbrick Researches, Inc., Dedham, Mass., Model P45AU and P66A (booster)). No distinction could be made between experimental results 33 Figure 5. 34 Circuit diagram of potentiostat with floating load resistor. C.A. E 3 Control amplifier Inverting input + 2 Noninverting input R. 2 Reference electrode W. 3 Working electrode C. 3 Counter electrode F G. 2 Function generator Detector Load resistor O 00 50 fit.)- 35 \5/ 01”! Figure 5 36 obtained with either instrument. Cell current was determined from the potential drop across the load resistor, 3L. To measure this potential drop two different recording devices were used depending on the time scale of the experiment. For eXperiments lasting less than about 1 sec a storage oscilloscope (Tektronix, Inc., Beverton, Ore., Type 564 with 2A63 (ver- tical) and 2B67 (horizontal) plug-in units) with Polaroid camera attachment (Tektronix Type C—12) was employed. For longer electrolyses an x—y recorder (Honeywell, Inc., San Diego, Calif., Model 520) was used. Both these record- ing devices have differential inputs which are well isolated from ground so that igL drop across the floating load resistor could be measured easily, and the circuit of Figure 5 functioned satisfactorily. However, for some early ex- periments a potentiometric recorder (Leeds and Northrup, Model G) was employed. This recorder cannot be operated differentially, and therefore for this recorder the circuit of Figure 5 had to be modified. A block diagram of the modi— fied circuit is given in Figure 6. Basically this circuit is the same as the one in Figure 5, but an additional amplifier is added to act as a current follower (39)(C.F.). This current follower maintains the potential of the working electrode at Virtual ground, and thereby eliminates intro- duction of 25L potentials into the control loop. At the same time the output of the current follower relative to ground potential is equal to ig_ , so that one side of the L detector can be grounded. Figure 6. 37 Circuit diagram of modified potentiostat. Control amplifier Inverting input Noninverting input Reference electrode Working electrode Counter electrode Function generator Detector Current follower Load resistor 38 C A} H\ Figure 6 39 The various electrochemical techniques used in this study (polarography, cyclic voltammetry, potentiostatic electrolysis and potential step-linear scan electrolysis) require different potential-time functions. For cyclic voltammetry and polarography this function (triangular wave) was obtained from a commercial function generator (Exact Electronics, Inc., Hillsboro, Ore., Model 255). For poten- tiostatic electrolysis a battery operated low voltage power supply was used to generate the gate. The function for the potential step—linear scan technique was obtained by sum- ming via a passive adding network a delayed triangular wave with a gate of Opposite polarity from the low voltage power supply. The triangular wave was obtained from the function generator, which contains an internal delay mechanism so that it is possible to trigger the function generator with the gate, making A easily variable. With this function gen- erator scan rates from 0.001 to 1,000 Hz with a maximum ampli- tude of 2 volts could be dialed directly. Polarograms were obtained with a Sargent Model XV Polarograph. ~For the spectrOphotometric measurements a Beckman Model DB was used. The cell arrangement is shown in Figure 7. The cell consists of a glass weighing bottle with a 60/12 9 joint on the top. The cell lid is made of Teflon and machined to fit this joint. Holes were drilled in the lid to allow insertion of the various cell components (Figure 7). Figure 7. 40 Cell arrangement. Reference electrode Scoop Hanging mercury drop electrode DrOpping mercury electrode Deoxygenator Counter electrode "211110001? .0 It‘s .0 I. .0 .0 42 The counter electrode consists of platinum wire (1 foot, number 26 gauge) wound around 6 mm soft glass tubing. One end of the platinum was sealed in the glass tubing, and electrical contact was made through mercury contained in the glass tubing. The reference electrode contained three separate sec- tions separated by 10 mm fine fritted glass discs. The left hand compartment (Figure 7) was a saturated calomel elec- trode, and the right hand compartment contained the solution being investigated. Because the latter solution contained perchlorate, lg sodium nitrate was used in the middle com— partment to prevent precipitation of potassium perchlorate. Liquid junction potentials obviously arise when different solutions are contacted in this manner. However, in this work only relative potential measurements were necessary. Presumably the liquid junction potential for a given solu- tion would remain constant during the course of an experi— ment. In fact, half—wave potentials for a given system never varied more than about 10 mv over the course of the entire investigation. The right hand compartment of the reference electrode could be disassembled at the 14/20 joint so that either a Luggin capillary or a tube closed with a 10 mm fritted glass disc could be used to contact the solu- tion. The Luggin capillary was constructed so that it could be positioned within less than 1 mm of the surface of the working electrode. In this way uncompensated ohmic poten- tial losses always were less than one ohm. 43 For experiments requiring a stationary electrode, a hanging mercury drop constructed in the manner described by Ross, DeMars and Shain (40) was used. Generally three mercury drops from the dropping mercury electrode (Sargent 8-29419 capillary, 2-5 sec) were collected in a scoop and suspended from the working electrode. The area of this electrode was 0.085 i 0.001 cm2 for all acid concentrations. The deoxygenator was of conventional design and could be raised above the solution to provide a nitrogen blanket over the solution while measurements were being made. All measurements were made in a constant temperature room at ambient temperature of 23-240C. Chemicals. Zone refined azobenzene (Litton Chemicals, Inc., Fernandina Beach, Fla.) was used without further treatment. mfAzotoluene was prepared by reduction of £7 nitrotoluene with zinc dust and sodium hydroxide (41). The product was purified by eluting it with petroleum ether (30-600) from an alumina column, followed by repeated re- crystallizations from ethanol. Melting point (54°C, uncor- rected) and infrared spectra agreed with the literature. Anal, Calcd. for C14H14N2: C, 79.97; H, 6.71; N, 13.32. Found: C, 80.06; H, 6.69; N, 13,36. meydrazotoluene was prepared by reduction of the mfazotoluene (36). Procedures. For electrochemical experiments procedures and preparation of solutions were identical with the descrip- tion of Schwarz and Shain (9,11) except that to avoid air 44 oxidation of the mfazotoluene, solvents were deaerated prior to dissolving the mfazotoluene. The solvent for all experiments was 50% (by weight) ethanol—water. As in the work of Schwarz and Shain no attempt was made to maintain constant ionic strength, because the salt effect in 50% ethanol is small (9). In general for spectrophotometric measurements the procedure of Carlin and 0dioso (36) was followed exactly. The one exception was that because the half-life of the re- action being studied was fairly small (ca. 70 sec.), the quenching procedure of Carlin and 0dioso (36) could not be employed. The procedure used consisted of rapid mixing of a solution of the mfhydrazotoluene in 50% ethanol with a solution of perchloric acid in 50% ethanol, followed by rapid transfer to the spectrophotometer cell. These opera- tions were easily completed within one half—life of the chemical reaction. Absorbance was then recorded directly as a function of time with a potentiometric recorder. Analysis of these absorbance-time data followed the pro- cedure of Carlin and 0dioso (36). RESULTS AND DISCUSSION Effects of Gelatin .Both azobenzene and hydrazobenzene adsorb on mercury (11,35,37). Most electrochemical studies of this system, however, have been performed at relatively low acid concen- tration (23—33) (pH = 2—13), where the extent of adsorption is small, and the rate of the benzidine rearrangement is slow with respect to duration of the experiment. Holleck and co-workers (30-32) investigated effects of surface active materials on the polarographic behavior of azobenzene and hydrazobenzene in 0.1M_perchloric acid. They concluded that gelatin became adsorbed on the electrode and inhibited the electron transfer reaction. Nygard (27) made similar studies using other electrochemical techniques, and his results generally agree with Holleck and co—workers. Using potentiostatic electrolyses, Schwarz and Shain (11), also reported an anomalous stirring effect which oc- curred at high acid concentration. This effect was evidenced by large periodic fluctuations in current-time curves; qualitatively the time at which this happened appeared to be.proportional to the rate of the benzidine rearrangement. Results of these previous investigations indicate that gelatin has two main effects. First, gelatin is preferen— tially adsorbed, which largely eliminates adsorption of depolarizer. Second, adsorbed gelatin inhibits the rate of 45 46 the heterogeneous electrode reaction. Even if the electron transfer rate is inhibited, however, it often may still be possible to select experimental conditions where the elec— tron transfer is reversible. Under such conditions the second effect of gelatin cited above should be unimportant, and the only effect of gelatin would be to eliminate adsorb- tion of depolarizer and reaction products. In this case measurements of homogeneous kinetic parameters on systems where adsorption is prevalent should be improved by using gelatin. Because if these conclusions are correct they would have not only important bearing on the present investi- gation, but on the study of chemical reactions coupled to electrode processes in general, a number of eXperiments were performed in an effort to define exactly the role of gelatin. Results of these experiments are described in the following paragraphs. The effects of gelatin on reduction of azobenzene were investigated using four techniques: conventional polaro- graphy, cyclic voltammetry, chronoamperometry, and potential step-linear scan electrolysis. Conventional polarographic studies were made with solutions of about 1 mg azobenzene with perchloric acid concentration from 0-5fl;t0 1.0M, and 0.00 to 0.05% gelatin. Results of these experiments are summarized in Table II. No meaningful data could be obtained in the absence of gelatin, because of extensive stirring phenomena which apparently are related to adsorption of de— polarizer, and which increase with increasing acid concentration 47 Table II. Polarographic data for the reduction of azoben- zene at various gelatin and acid concentrations. CS,Q§, [H+),M. Gelatin,% iL,u amp iL/Cg x103 Ej/z v 1.000 0.500 0.00 6.17 --- -—- 0.980 0.490 0.01 5.58 5.70 0.025 0.962 0.481 0.02 5.39 5.63 0.025 0.943 0.472 0.03 5.28 5.61 0.024 0.926 0.463 0.04 5.16 5.57 0.024 0.909 0.455 0.05 5.10 5.62 0.024 1.000 1.00 0.00 --- —-- --- 0.980 0.98 0.01 5.76 5.87 0.059 0.962 0.96 0.02 5.52 5.76 0.062 0.942 0.94 0.03 5.39 5.73 0.062 0.926 0.93 0.04 5.22 5.64 0.061 0.909 0.91 0.05 5.25 5.77 0.061 1.000 1.50 0.00 --- --- --- 0.980 1.47 0.01 -—- —-- --- 0.962 1.44 0.02 5.64 5.86 0.095 0.942 1.41 0.03 5.45 5.78 0.096 0.926 1.39 0.04 5.40 5.82 0.096 0.909 1.36 0.05 5.25 5.77 0.095 1.000 2.00 0.00 --- --— —-- 0.980 1.96 0.01 --- --- --- 0.962 1.93 0.02 5.70 5.92 --- 0.943 1.89 0.03 5.57 5.92 0.132 0.926 1.85 0.04 5.45 5.90 --- 0.909 1.82 0.05 5.34 5.87 --- aHalf-wave potentials are versus S.C.E. 48 (see Figure 8). However, for higher gelatin concentrations the normalized limiting current, iL/Cg, is a constant within experimental error for all acid concentrations. Moreover, values of El/z appear to be independent of gelatin concen— tration. This last fact implies that even if gelatin does decrease the rate of the electron transfer reaction, this reaction is still rapid enough to be reversible under polaro- graphic conditions. Cyclic voltammetry also was used to study the effect of gelatin on anodic and cathodic peak currents in approx- imately 0.5M perchloric acid. With cyclic voltammetry rate constants for a succeeding chemical reaction can be deter- mined from the ratio of anodic to cathodic peak currents (10). In addition rate constants for rearrangement of hydrazobenzene in 0.5M perchloric acid are accurately known (9). Thus, it is possible to measure rate constants with cyclic voltammetry in solutions both with and without gela- tin present, and to compare these rate constants with the known values. Results of these experiments, which are listed in Table III, show very good agreement between the known and measured rate constants. Normalized cathodic peak cur- rent, ip/Cg, decreases slightly when gelatin is added, pre- sumably because preferentially adsorbed gelatin reduces the amount of weakly adsorbed azobenzene (37). At higher acid strengths than 0.5M_it is impossible to make kinetic measure- ments without gelatin present because of the stirring phenom— ena cited in relation to the polarographic experiments (Figure 9). Figure 8. 49 Conventional polarographic current—potential curves showing effects of gelatin on reduction of azobenzene. 1mg azobenzene, 0.5M perchloric acid. A, no gelatin; B, 0.02% gelatin; 1mg azobenzene, 1.0M perchloric acid. C, no gelatin; D, 0.02% gelatin. 50 12.0—- C 9.0—- Q. E f0 :1 .5 C.‘ 0) H A H :3 U 6.0-— :__—"——"——_—— g B 3.0-— 0." l |—- I l l 0.2 0.1 0.0 -0.1 —0.2 Potential y§_ S.C.E., volt Figure 8 Table III. c 1.2M. * OI 1.000 0.980 0.962 0.943 0.909 51 Comparison of rate constants for rearrangement of hydrazobenzene determined with and without gelatin present. [H1],M_ Gelatin,% .iEfiEE. k_sec71a ksec?1 0.500 0.00 0.023 0.204 0.204 0.490 0.01 0.022 0.192 0.256 0.481 0.02 0.022 0.181 0.187 0.472 0.03 0.022 0.172 0.180 0.455 0.05 0.022 0.156 0.160 aRate constants determined by Schwarz and Shain in the absence of gelatin (9). bRate constants determined with cyclic voltammetry using ratio of anodic to cathodic peak currents. b 52 Figure 9. Stationary electrode polarograms showing effects of gelatin on reduction of EM azobenzene (scan rate, 30 mv sec 1 A, 0.5-2.0M_perchloric acid, 0.02% gelatin. B, 1.0M_perchloric acid, no gelatin. C, 2.0M perchloric acid, no gelatin. Current 53 Potential Figure 9 54 The effect of gelatin on diffusion coefficients was determined by calculating diffusion coefficients from potentiostatic electrolysis data. This technique was chosen because adsorption effects of depolarizer are minimized and thus the effect of gelatin adsorbed on the electrode sur- face could be evaluated. Solutions of about 1mg azobenzene, approximately 0.5 and 2.0M_perchloric acid, and from 0.00 to 0.05% gelatin were employed. Data were analyzed by the conventional plot of i_versus lfift, and results are listed in Table IV. Clearly, from these data the measured dif— fusion coefficient is not affected by the presence of gelatin. There may, however, be a small influence of ionic strength since the values of D are slightly higher in 2.0g 0 perchloric acid. The average diffusion coefficient of all values in Table V is 3.3 x 10-6 cm2 sec-1. A value of 3.4 x 10-6 cm2 sec_1 was determined by Schwarz and Shain (9) for measurements in the absence of gelatin. For potentiostatic experiments, the same stirring ef- fect reported by Schwarz and Shain (see Figure 10), and mentioned above in connection with polarographic and cyclic voltammetric experiments, was observed. As in the case of polarography and cyclic voltammetry this effect is elimin- ated for any acid concentration when at least 0.02% gelatin is present. Very recently, Wopschall and Shain (37) treated theor- etically the effects of adsorption processes on single-scan and cyclic stationary electrode polarograms. Based on their Te (2 55 Table IV. Diffusion coefficients of azobenzene for dif- ferent acid and gelatin concentrations. b C8,gfl_ [H+],fl' Gelatin,% Slopea Egzxsiggi 1.000 0.500 0.00 16.3 3.1 i 0.1 0.980 0.500 0.01 16.0 3.1 i 0.1 0.943 0.500 0.03 15.7 3.2 i 0.1 0.909 0.500 0.05 15.1 3.2 i 0.1 1.000 2.00 0.00 16.9 3.3 .t 0.1 0.980 2.00 0.01 16.7 3.4 i 0.1 0.943 2.00 0.03 16.3 3.5 i 0.1 0.909 2.00 0.05 15.3 3.3 i 0.1 aSlope determined from plot of i_versus lflft. bError level is average deviation of three eXperiments. 56 .cflumamm on .Uflom UHHOHQUHmd.flo.H .m .5538 4.8.0 .38 3823mm mfmumé .4 .mcwmcmnomm mo coauodcmu co CHDmHmm mo muommwo mCHBOLm mm>Hdo UHHuoEOHwQEmocoHLU .OH mudmflm 57 OH mudmflm mane V 0mm H.o dme n 93 quaIInD 58 theoretical results and the eXperimental results already discussed, the azobenzene system appears to fit a model where in the absence of gelatin weak adsorption of both azobenzene and hydrazobenzene occurs. However, gelatin apparently is preferentially adsorbed, and therefore in the presence of gelatin adsorption of azobenzene and hydrazo- benzene is essentially eliminated. Also gelatin slows the rate of the electron transfer, but the effect is not great, and conditions easily can be selected where the process remains reversible. Because the potential step—linear scan method is a combination of cyclic voltammetry and potentiostatic elec- trolysis, the effects of gelatin on this method should be a combination of effects already discussed for the latter two methods. Experimental results, which indicate that this is the case, are discussed in the following section along with applications of the potential step-linear scan theory. Reduction of Azobenzene To check the theoretical calculations for the potential step-linear scan method, the azobenzene system can be used because of the extensive electrochemical data already avail— able (9,11,35). Experimental conditions were arbitrarily selected identical with those of Schwarz and Shain (9). In general the azobenzene system was found to behave exactly as described by Schwarz and Shain (9,11). The one exception 59 is that gelatin does tend to eliminate the anomalous stir— ring behavior Schwarz and Shain observed when electrolysis times are longer than the half-life of the chemical reaction. As shown below, the presence of gelatin does not have a significant effect on rate constants measured by potential step-linear scan. In Figure 11 experimental and theoretical potential step-linear scan curves for azobenzene are compared. The experimental curve (solid line) actually corresponds to two different experiments, one in which gelatin was employed (0.02%), and another in which gelatin was not used. These two curves are within experimental error and therefore are represented by a single line. The theoretical currents (circles) were calculated from Equation 22 using 3; = 10.4, n'= 2.0, c; = 1.0mM_and the diffusion coefficient determined by Schwarz and Shain (9) of 3.4 x 10-6 cm2 sec—1. To obtain the comparison between theoretical and eXperimental curves the value of k/a_ in the theoretical calculations was varied to give the best fit. Clearly the agreement between theory and experiment is excellent. However, it might be argued that the reason for this agreement is the adjustable parameter, k/aj.which was selected to give the best agree- ment. To answer this question it is necessary to determine if the value of k_which gives best agreement with theory is the correct rate constant for the chemical reaction. There- fore, an evaluation of rate constants determined by the po- tential step-linear scan theory is discussed next. 60 Figure 11. Comparison of theory and experiment for potential step—linear scan method. 0, theory (g,_\ = 10.4, 5/3 = 0.03) -——, experimental for reduction of mM azobenzene in 0.5M perchloric acid. Scan_ rate, 94 mv sec ; El/z, 0.024 v vs S .C. E. ; A, 0.085 cm2; and A, .42 sec. 61 20F- 15—- 10- uams> U1 Current, 0 1 ~10 ._ -15 _. _ 'J ‘0 Q fi C: C‘ 0 Q J J l ‘1 C C 0 1 7\ 2 Time, sec Figure 11 62 The fact that rate constants determined with the aid of the working curves discussed above (Figure 4) agree with rate constants determined by an independent electrochemical method is illustrated in Figure 12*. In Figure 12 circles represent rate constants determined with the potential step- linear scan theory developed above. Each circle corresponds to measurements both with and without gelatin; results in the two cases are within experimental error. Nevertheless with gelatin present much larger values of.A could be used than in the absence of gelatin. The squares of Figure 12 represent rate constants determined by Schwarz and Shain (9). With the potential step-linear scan method it was possible to cover as wide a range of rate constants as Schwarz and Shain reported for the squarewave technique (9). Moreover, the range is considerably greater (in the direc- tion of large rate constants) than Schwarz and Shain studied with their potential step-linear scan theory (11) (the largest rate constant they report is log(k) - 0.4). Although not stated explicitly by Schwarz and Shain the reason for this upper limit was no doubt set by the approximations of their theory. Thus, for them to measure larger rate con- stants, and still satisfy the assumptions of their theory, would have required prohibitively large scan rates where effects of adsorption, double layer charging, and charge transfer kinetics become dominant. *At high acid concentrations in 50% ethanol the Hammett acidity function, Ho, must be used as a measure of acidity, rather than the molarity of the acid. Values of H0 in Figure 12 were obtained from the paper of Schwarz and Shain (9). 63 -Figure 12. Variation of k_with ED for the benzidine rearrangement of hydrazobenzene. CL potential step-linear scan method. CL squarewave electrolysis (9). log (k) 64 65 Reduction of mfiAzotoluene The potential step—linear scan method also was used to study the reduction of mfazotoluene. Qualitatively this system behaves the same as azobenzene. Some differences were Observed, however. For example, the anomalous behavior reported by Schwarz and Shain (11) for long electrolysis times was even more pronounced than with azobenzene. As with azobenzene :it was found that gelatin minimized this effect without affecting the value of measured rate constants. Most of the data reported for grazotoluene were obtained with 0.01% gelatin. Another difference between.m7azotolu- ene and azobenzene is reversibility of the electrode reac- tion. Thus, under identical experimental conditions (with— out gelatin) the apparent heterogeneous rate constant for azobenzene is a factor of 2 or 3 larger than for grazo- toluene. This effect presumably is related to structural differences in the two compounds, and is discussed in more detail in a later section. The final major difference be— tween the two compounds is the rate of the benzidine rear- rangement. At identical acid concentrations, mfazotoluene rearranges roughly 5 times faster than azobenzene. This effect is well known and explained bythe inductive influence of the.methyl groups. Rate constants determined for rearrangement of mfhydrazo- toluene are summarized in Table V, and are plotted against acid concentration* in Figure 13 (circles). The slope of *- For these acid concentrations it is not necessary to use the Hammett acidity function. 66 Table V. Rate constants for rearrangement of mfhydrazo- toluene. Acid Conc.a,ML kb, sec"1 kd, sec-1 0.05 0.0067; 0.0059 -- 0.06 o .0126C -— 0.07 0.0133 -- 0.08 0.0177 0.021 i .003 0.10 0.035 0.036.i .003 0.20 -- 0.138 i .004 0.30 -- 0.329 i .007 0.40 —- 0.769 t .017 aFor rate constants in the second column the acid was hydro— chloric (however, see footnote c); in the third column the acid was perchloric. bExtrapolated values of Carlin and 0dioso (36). cFor spectrophotometric measurements under electrochemical conditions 0.010 and 0.015 sec‘1 were obtained. dRate constants measured by potential step-linear scan. Numbers in parentheses are average deviations of at least 12 experiments. 67 Figure 13. Variation of k_with concentration of perchloric acid for rearrangement of mfhydrazotoluene. CK potential step—linear scan method. CL spectrophotometric technique (36). *, determined spectrophotometrically under electrochemical conditions. 68 log (k) l | -0.5 -1.0 log [H+] Figure 13 69 the straight line drawn in Figure 13 is 2.16 which is con— sistent with the well—known second order dependence of benzidine rearrangements on acid concentration. Comparison of rate constants. Rate constants for rear- rangement of mfhydrazotoluene have been determined spectro- photometrically by Carlin and 0dioso (36), and it is pos- sible to compare these results with the electrochemical results reported above. Unfortunately the classical measure- ments were performed under different experimental conditions where the rate of the reaction is slower (lower temperature and lower dielectric constant solvent—~95% ethanol rather than the 50%). Nevertheless the dependence of rate con— stant on dielectric constant and temperature is known (36, 42), and therefore it is possible to extrapolate the rate constants of Carlin and 0dioso to the electrochemical experimental conditions. This same procedure was used by Reilley and co-workers (35), who for one specific acid con- centration found reasonably good agreement between extra- polated classical and electrochemical measurements. Rate constants of Carlin and 0dioso extrapolated to conditions used in the electrochemical experiments also are included in Figure 13 (squares) and Table V. The agreement between the two approaches is excellent, and appears to justify the extrapolation. In spite of this a number of criticisms can be raised against the extrapolation procedure, and it might be argued that the agreement of Figure 13 is at least partly 70 fortuitous. To test this possibility the measurements of Carlin and 0dioso were repeated for one acid concentration, but for the same solvent and other experimental conditions employed for the electrochemical measurements (see discus— sion under Experimental). The rate constant determined in this way is identical within experimental error with the extrapolated value (see Figure 13 and Table V). Thus, the extrapolation is valid, and one concludes that at least for the mechanism and time scale considered here homogeneous rate constants can be measured electrochemically with com— plete confidence. REDUCTION OF AZOBENZENE IN NIN-DIMETHYLFORMAMIDE After using azobenzene compounds to evaluate the poten- tial step—linear scan theory, it was of interest to attempt to determine the mechanism of the electrode reaction itself. Studies with the potential step-linear scan technique all were performed under conditions where the electrode reac— tion was completely reversible. However, it is possible to make electrochemical measurements under conditions where the electrode reaction is perturbed away from equilibrium, and in this way kinetics of the electrode reaction can be measured. Preliminary heterogeneous kinetic studies in water indicated a 2-electron reduction involving hydrogen ions. Because of the complications resulting from hydrogen ion these aqueous data were not readily interpretable, and it was concluded that studies of the reduction mechanism of azobenzene in an aprotic solvent might profitably aid inter- pretation of aqueous data. N,N-Dimethylformamide (DMF) was chosen because a considerable number of electrochemical studies have been made in this solvent. Although DMF is not an aprotic solvent, studies (43) have shown the proton availability is small. Results of the non-aqueous studies are discussed first, and the aqueous work constitutes the final part of this thesis. During the course of this investigation Aylward, Garrett, and Sharp (44) published a report on the reduction of 71 72 azobenzene in DMF with 0.1M tetraethylammonium perchlorate supporting electrolyte. These authors used conventional d.c. and a.c. polarography. In addition, controlled poten- tial electrolysis was used to determine n_values, and ESR spectroscopy was used to examine reduction products. Their results show that azobenzene is reduced in two one electron steps. The first electron addition occurs at a value of EJ/z of —1.81 v and the second at EJ/z of —2.29 v versus a silver—silver nitrate reference electrode. An ESR spectroscopic study showed that the product of the first electron transfer is the monoanion of azobenzene, and the product of the second electron transfer is diamagnetic. Alternating current polarography and faradaic impedance measurements indicated that the first electron transfer is rapid with a heterogeneous rate constant of 0.51 i 0.05 cm sec-1, and the second electron transfer is irreversible. They postulated that the product of the second step is the dianion of azobenzene. The diffusion coefficient for azo- benzene in DMF was found to be 7.7 i 0.3 x 10.6 cm2 sec—1. The independent studies of azobenzene in DMF with cyclic voltammetry reported in this thesis are in essential agreement with the work of Aylward, Garrett, and Sharp. The work of this investigation actually complements the re- sults of Aylward.and co—workers, and where appropriate, com— parisons between the two studies are made. Although the original objective of studying the reduc- tion of azobenzene in aprotic solvent was to aid interpretation 73 of aqueous data, because of the unambiguous nature of the mechanism found in DMF, these studies also offered the opportunity of studying effects of molecular structure on reactivity. A number of correlations between half-wave potential (thermodynamic correlations) and structure have been reported in the literature. In spite of this apparently no correlations between reactivity--i.e., heterogeneous rate constant of the electrode reaction—-and structure have been made. By studying the reduction kinetics in DMF of a series of substituted azobenzene compounds, some correla- tions between reactivity and structure were possible. Al- though preliminary in nature results of these studies are of interest and also are described. EXPERIMENTAL Instrumentation Circuits and cells were identical with those already discussed in connection with experimental applications of the potential step—linear scan theory. Materials Solvent. Fisher certified reagent grade N,N-dimethyl- formamide (DMF) was used as solvent without further purifica- 1 tion. The specific conductivity was 1.4 x 10'6 (ohm cm) which is about one order of magnitude larger than values reported by Thomas and Rochow (45) for "Pure" DMF. Con— ductivity measurements were made for solutions containing 0.1, 0.2, and 0.3M tetraethylammonium perchlorate (sup- porting electrolyte) to obtain an estimate of the uncompen— sated resistance. The resulting conductivities were 0.0056, 0.0093 and 0.0126 (ohm cm)‘1 respectively. Chemicals. 4,4‘—Diethylazobenzene, 2,2‘—, 3,3'— and 4,4'- dimethylazobenzene were prepared by reduction of the corresponding nitro compounds with sodium hydroxide, methanol and zinc dust (41). prthylnitrobenzene was prepared by mononitration of ethylbenzene (46). The fraction boiling at 126°C and 13 mm (46) was used for the reduction. 74 75 4,4'—Dibromoazobenzene, 4,4'-dichloroazobenzene and 4,4'—dinitroazobenzene were prepared from the corresponding peaniline by oxidation with silver iodide dibenzoate in anhydrous benzene (47), followed by repeated recrystalliza- tion from ethanol and acetone. 4,4'-Bis(acetamido)azobenzene and 4,4‘—diaminoazoben- zene were prepared from pjaminoacetanilide (48). 1,1'-Azonaphthalene was obtained from K and K Labora- tories Inc., Plainsview, N.Y.—Hollywood, Calif. and 4,4'— azobis(N,N—dimethylaniline) was obtained from Eastman Organic Chemicals, Rochester, N.Y. The observed melting points for the various compounds are listed in Table VI. Tetraethylammonium perchlorate was prepared by meta- thesis of tetraethylammonium bromide with sodium perchlorate (49). The product was recrystallized six times from dis- t:illed water and dried at 660C. Procedures Solutions. Solutions were prepared by dissolving ac- curately weighed samples of the apprOpriate azo-compound and tetraethylammonium perchlorate in enough DMF for a final solution volume of 100 ml. These solutions were generally about 4 x 10'4M in azo—compound, and 0.3M_in tetraethyl- ammonium perchlorate. 76 Table VI. Comparison of observed and literature values for melting points of substituted azobenzene compounds. Compound M.P.a,0C M.P b,°C Azobenzene 68 68 (24) 4,4'-Bis(acetamido)azobenzene 288 288-293dec.(48) 4,4'—Diaminoazobenzene 238-239 238—241dec.(48) 4,4'-Dibromoazobenzene 205 203-205 (47) 4,4'-Dichloroazobenzene 184 184-185 (47) 4,4'—Diethylazobenzene 59—60 59-60 (61) 2,2'-Dimethylazobenzene 55-56 55 (62) 3,3'-Dimethylazobenzene 54-55 55 (63) 4,4'-Dimethy1azobenzene 141-142 142 (47) 4,4'-Dinitroazobenzene 224—225 223-225 (47) aObserved melting point. bLiterature melting point, reference in parentheses. 77 Evaluation of Diffusion Coefficients. Diffusion coefficients were evaluated from cyclic voltammetry experi- ments with scan rates sufficiently slow that the couple under investigation was reversible (see diagnostic criteria for reversibility in reference (10)). Under this condi- tion the expression (10) relating diffusion coefficient to the cathodic peak current, ip, is i ’J—Do = E}— . 268 n?”2 A CBIJv For these experiments A_ and y_ were 0.0621 cm2 and 0.630 volts/sec respectively, and the bulk concentration of azo- compound was approximately 1.03M. The supporting electro— lyte was 0.3M_tetraethylammonium perchlorate. Evaluation of E1/2' Values of El/z for azo—compounds also were determined with cyclic voltammetry. Again, scan rates were such that the couple was reversible. Under this condition the value of El/ is the potential at 85.14% of /2 the cathodic peak current (10). Measurement of Heterogeneous Rate Constants. Hetero— geneous rate constants, kg, for substituted azobenzene com- pounds also were determined from cyclic voltammetric data. This approach involves using scan rates sufficiently large that the electrode reaction is perturbed from equilibrium, which is indicated by an overpotential between cathodic and anodic peak potentials (AE ). Recent theoretical calculations 78 (14) provide the quantitative relationship between égp' scan rate, and the rate constant, ks. These theoretical relation— ships are presented in the form of a working curve, which is a plot of n_x éfip against a dimensionless parameter, Q. The parameter y_has the value 19gsfifigfig , where y is (Do/DR)1/2, __is the transfer coefficient, and other terms were defined earlier. Thus, to use this approach to measure ks, a large plot of the working curve was constructed, and this plot was used to relate experimental égp values to y, Since scan rate was accurately known, values of ks could be calculated from g, provided 1 and g were knbwn. Actually, for the large organic molecules studied here, with both oxidized and reduced forms soluble in solution, D0 and DR are nearly the same. Moreover, g is usually about 0.5, and therefore, it is reasonable to assume that the quantity 1? is nearly unity. This approximation was made in all measurements of kg. The working curve described above was calculated for g_= 0.5; for different values of g_a different working curve results. Fortunately, however, for large values of g_all working curves are independent of g, For example, for g. 0.1 there is about a 20% difference Of.é§p for_g between 0.3 and 0.7, but for larger 1 the variation rapidly becomes less. Therefore, to minimize this source of error, scan rates were selected to correspond to values of y_always larger than 0.5,and in this way errors resulting from in— accurate knowledge of g were at most 5% (14). 79 For all experiments the initial potential was at least 160 mv anodic of gl/Z, and switching potential was about 140 mv cathodic of El/z' Switching potential is not criti- cal as long as it is reasonably near the 141/n mv used in theoretical calculations of the working curve (14). Although the use of cyclic voltammetry to measure heterogeneous rate constants in the manner just described is very simple and convenient, a serious source of error is uncompensated ohmic potential losses ($3 drop). This is because of the fact that uncompensated 23 drop causes "ap- parent" overpotentials which vary with scan rate in approxi- mately the same way as activation overpotentials (38). Thus, it is essential that the uncompensated £3 drOp be small and known. To minimize uncompensated resistance, the 3-electrode circuits already discussed were used in con- junction with a Luggin capillary reference probe which was placed in close proximity with the working electrode (< 1mm). In addition, a relatively high concentration of supporting electrolyte was used to increase conductivity (0.3fl;tetra- ethylammonium perchlorate), and a relatively low concentra— tion of depolarizer (4.0 x 10‘4M) was used to minimize cell current. Under these conditions the largest cell currents encountered were about 0.1 ma, and based on the conductivity measurements, and the discussion of Booman and Holbrook (50), the maximum uncompensated ig_loss was about 1 mv. RESULTS AND DISCUSS ION Cyclic voltammetric data for azobenzene confirm the results of Aylward and co-workers. A typical cyclic polaro- gram for reduction of azobenzene is shown in Figure 14. The first wave corresponds to a one electron reversible electron transfer as is evident from the peak potential separation of approximately 60 mv (10) and the value of about 60 mv for the quantity (EP/z - Ep) (10). The second wave corresponds to an irreversible one electron addition. The peak poten- tial separation for the two cathodic waves is about 500 mv. Actually, cyclic polarograms of 4,4'—, 3,3'- and 2,2'- dimethylazobenzene, 4,4'-dichloroazobenzene, 4,4'-dibrom- azobenzene, 4,4'-diethylazobenzene and l,1'—azonaphthalene are identical in all important aspects with curves for azo- benzene. Thus, curves of Figure 14 can be considered rep— resentative of all of the above compounds. Based on these results it was concluded that all of these compounds are reduced by the same mechanism as azobenzene——i.e., the mechanism of Aylward and co-workers. The reduction of 4,4'-dinitroazobenzene is complicated because nitro groups are reduced at about the same potential as the azo-linkage. Because in DMF the nitro reduction also occurs in steps, it is difficult to determine exactly which wave corresponds to reduction of the azo group. Furthermore, difficult base line corrections would have to be applied 80 81 Figure 14. Cyclic stationary electrode polarogram for reduction of mM azobenzene in 0.3M tetraethyl- ammonium perchlorate and N, N— —dimethylformamide (scan rate, 40 v sec 1). 82 Current -1.4 -1.6 —1.8 —2.0 -2.2 Potential y§_S.C.E., volt Figure 14 83 to obtain quantitative information. For these reasons this system was not investigated further. 4,4'-Azobis(N,N-dimethylaniline), 4,4'-diamino- and 4,4'-bis(acetamiddazobenzene also show deviation from the results obtained for azobenzene. Two waves are observed in the cyclic voltammetry, current potential curves. However, the first wave is broader [(E - E ) is about 80 mv], and T} -p/ 2 -p .,_ the anodic current is considerably reduced even at very é large scan rates (ca. 100 volts/sec). A second wave also i is observed at more cathodic potentials, however, it occurs 9 at a potential only about 150 mv cathodic of the first. A mechanism where the product of the first electron transfer disproportionates to give starting material and the product reduced at the second wave is consistent with these results, but no detailed study was made. Evaluation of Diffusion Coefficients ,To calculate values of 58 from the parameter g_(see discussion under Experimental) it is necessary to know accurately the diffusion coefficient, 29, for the oxidized form of depolarizer. Therefore, diffusion coefficients in DMF were determined for all azobenzene compounds on which heterogeneous kinetic measurements were made. Results of these measurements are summarized in Table VII. Comparison of Measured Diffusion Coefficients with §tokes—Einstein Diffusion Theory. It is interesting to note 84 Table VII. Diffusion coefficients of substituted azobenzene compounds in N,N-dimethylformamide. , a i l ‘- D0 X 106 —1/3 Compound p C* mM cm2 sec_i (M.W.) )u am 0’-- Azobenzene 36.8 1.000 7.8 i 0.2b 0.177 1,1'-Azonaphthalene 34.3 1.009 6.7 i 0.2 0.154 4,4'-Dibromoazobenzene 32.8 1.000 6.2 i 0.2 0.143 4,4'-Dichloroazobenzene 34.3 1.010 6.7 i 0.2 0.158 4,4'-Diethy1azobenzene 33.9 0.981 6.9 0.2 0.161 2,2'-Dimethy1azobenzene 35.5 0.990 7.3 0.2 0.168 ++ 1+ H 3,3'-Dimethy1azobenzene 34.8 0.970 7.4 0.2 0.168 H- 4,4'-Dimethy1azobenzene 35.0 0.985 7.3 0.2 0.168 a O I O O I Error level is average deViation of Six eXperiments. bAylward and co-workers (44) report a diffusion coefficient of 7.7 i 0.3 x 10"6 cm2 sec‘1_for azobenzene. 85 that diffusion coefficients of Table VII depend on molecular weight. Thus, these data provide the interesting possibil- ity of evaluating the Stokes-Einstein diffusion equation for calculating diffusion coefficients of molecular solutes. The Stokes-Einstein equation (13) is gr. 1 .A 6wnr D: where Q_is diffusion coefficient, 3 viscosity coefficient of the solvent, £_radius of the molecule, and A_Avogadro's number. If the molecules are large spheres which do not polymerize or undergo appreciable solvation, and the molar volume in solution is the same as in the pure state, the radius can be calculated from the molecular weight, grfl3, and the density, d, of the substance in the pure state by the relation Combination of these equations and evaluation of constant terms (T = 2980K) results in -7 1/3 D : 2.96 x 10 (d) cmz sec—1 n(M.W.)1f3 The dependence of Q_on (M.W.)-1/3 is illustrated in Figure 15. The slope of the line in Figure 15 has a value of 4.9 x 10-5. wHowever, based on a viscosity coefficient for pure DMF of 0.00802 dyne sec cm.2 (51) and the assumption that density for the azobenzene compounds is 1.0, the 5 theoretical slope is 3.70 x 10' The assumption of unity 86 Figure 15. Variation of diffusion coefficient of substituted azobenzene compounds with molecular weight (error level, i 0.2 x 10"6 cmzsec‘1). 87 7.8- 6.2?- .16 .17 0.18 -1 (M.W.) /3 .15 .14 Figure 15 88 for density is reasonable because of the exponent (1/3) as- sociated with density. Thus, errors in the theoretical slope resulting from inaccurate values of d_are at most about 6%. This error is smaller than the difference be— tween theoretical and measured lepes. Experimental dif- fusion coefficients were determined in solutions containing 0.3g_tetraethylammonium perchlorate. The Stokes-Einstein diffusion equation in the above form is only approximate (see above discussion) and generally applies at infinite dilution. These facts probably explain the discrepancy between theoretical and observed slopes. Effect of Double Layer Structure on.Heterogeneous Rate Constants The heterogeneous rate constants measured experimen- tally with cyclic voltammetry in the manner already discus- sed are apparent rate constants, because of the well-known influence of electrical double-layer structure on hetero- geneous rate constants (52). Hence, for any of the con— clusions made below between structure and reactivity to be meaningful, it is essential to evaluate effects of double- layer structure on the measured rate constants. Fortunately, for relative comparisons of the type discussed below, it is only necessary that double-layer corrections be constant for the series of rate constants. The discussion which follows demonstrates that these corrections are constant, and therefore no formal attempt to correct the apparent rate constants was made. 89 Electrical double—layer structure for tetraalkyl- ammonium iodides at mercury—water interface has been investi— gated by Devanathan and Fernando (53). Their results for tetraethylammonium iodide indicate that the charge of the interface is relatively constant (15 to 17 u coulombs cm—Z) over the potential range considered in this investigation (-1.2 to -1.6 y_versus SCE). Double-layer effects in DMF have been investigated by Minc and co-workers (54). Although they used lithium chloride and cesium iodide as supporting electrolyte, Frumkin (55) and Devanathan and Fernando (53) have shown that in water tetraalkylammonium iodide and cesium iodide cause very similar double-layer effects. At negative potentials anion contributions to double-layer structure are small because of repulsion from the electrode, and thus changing from iodide to perchlorate,which was used in this study, would result in only slight differences. Thus, by assuming that double-layer effects for tetra- ethylammonium idoide and cesium idoide are about the same in DMF, and applying Gouy-Chapman theory (applicability of this theory to non-aqueous solvents has been discussed by Grahame (56)) to evaluate these effects, the corrections which would be applied to apparent rate constants were found to be constant within about 5%, i.e.-within experimental error . 90 Applicability of the Hammett Equation If changes in free energies of several related equi- libria are proportional to the change in some property of the substituents (such as electron withdrawing ability or resonance effects) a linear free-energy relationship results. Hammett has shown that a proportionality of this type de- scribes the changes produced by meta and para substituents on equilibria involving phenyl-substituted compounds (57). For benzene derivatives, X-C6H4-R, which have a po- larographically active group, R, and substituent,-X, in the meta or para position the Hammett equation for shifts in half wave potential with substituents (58) has the form AEl/z = p0 where g_is the Hammett substituent constant (57). Values of substituent constants recently have been tabulated by Jaffe (59), and these values were used in comparisons below. ‘9 is the reaction constant (expressed in volts) which re- flects the sensitivity of the electrode process to the effects of substituents. It should be noted that the linear free energy relationship which is described by the above equation corresponds to free energy change for the overall electrode process i.e., free energy change between reactants and products. An interesting fact about the generality of the Hammett equation is that it also has been used to correlate rates of reactions with substituent effects. This fact has made 91 possible the use of the Hammett equation in studying re— action mechanisms and effects of structure on reactivity. Thus, one might expect that the standard heterogeneous rate constants determined for a series of molecules of the type described above also would give a linear correlation with substituent constants. The equation relating Es and the 1 free energy of activation, AG , is kg = c eXp(-AGi/RT) where Q.is a constant and §_and T_have their usual meaning. Hence, a modified Hammett equation relating kg and g has the form I! log ks : p0 so where k and k are the heterogeneous rate constants for -—s '—so the substituted and unsubstituted compounds reSpectively, and g and p_are defined above. Implicit in the derivation of this equation is the assumption that the reactants are in equilibrium with activated complex; therefore, if a linear free energy relation exists between ks and g, the free energy change between reactants and activated complex depends on substituents. Variation of Half-Wave Potentials with Structure In Table VIII measured values of EJ/z for substituted azobenzene compounds are listed. These data are plotted versus the corresponding Hammett g values in Figure 16. 92 Table VIII. EJ/z and ks values for azobenzene compounds in N,N-dimethylformamide. V' AE k b a — nx _ volts - -p -s compound E1/2 sec"1 mv ¢ (cm sec'”I Azobenzene 1.420 29.1 77 1.41 0.25 43.5 79 1.25 0.25' 1,1'-Azonaphthalene 1.200 30.0 80 1.19 0.18 42.3 84 0.99 0.18 4,4'-Dibromoazobenzene 1.256 30.0 70 2.40 0.35 42.3 73 1.85 0.34 4,4'-Dichloroazobenzene 1.265 30.0 70 2.40 0.37 42.3 72 2.00 0.37 4,4'—Diethy1azobenzene 1.511 30.0 83 1.03 0.16 42.3 86 0.91 0.17 2,2'-Dimethylazobenzene 1.463 30.0 89 0.82 0.13 42.3 93 0.70 0.14 3,3'-Dimethy1azobenzene 1.468 28.5 77 1.42 0.23 42.7 80 1.19 0.23 4,4'-Dimethylazobenzene 1.534 27.5 86 0.91 0.14 41.4 90 0.79 0.15 aEJ/z values versus aqueous SCE, reproducibility i bError level of i10% (see discussion in text). 10 mv. 93 Figure 16. Variation of EJ/z with Hammett g_value for substituted azobenzene com ounds (slope, 0.36 volts; Ell/2 is S.C.E. . 94 O 0.0 2 IO _ 4 l0 _ +, C _ _ _ 6 5 4 3 2 1 1. 1 1. 1 muao> .«\Hmn Figure 16 95 The points in Figure 16 define a straight line with slope (i.e., reaction constant, p) of 0.36 i 0.01 volts. Sub- stituted azobenzene compounds have been studied in aqueous solutions and correlations of this type also have been made (58). Values of Q_which were obtained in these correlations are considerably lower (0.08 at pH < 3.0 and 0.16 at pH = 5-8) than the value obtained above. However, in aqueous solutions the reduction of azobenzene compounds follows a different mechanism than in DMF. The mechanism in aqueous solutions involves hydrogen ions and is'not known in detail, therefore it may be that half-wave potentials are not directly related to free energy changes. For these reasons a direct comparison between the values oflg reported here and those obtained by other workers is not possible. How- ever, for polarographic studies of substituted nitrobenzene compounds in DMF with 0.hfl tetraethylammonium iodide (60) a reaction constant of 0.42 was determined from correlations of half-wave potentials for the first reduction step forma- tion of nitrobenzene radical anion) with Hammett g values. Because of the similarity of azobenzene and nitrobenzene compounds, and since the reduction mechanisms appear to be similar, the observed agreement between reaction constants for these series of compounds is reasonable. Variation of Heterogeneous Rate Constants with Structure Measured values of kg for azobenzene compounds are listed ianable VIII. These rate constants are accurate 96 within about i10%, even though measured rate constants show an average deviation of about i3% (6 runs), because of a combination of errors from g, 20' etc. Figure 17 is a plot of log ks versus Hammett g values (59) for substi- tuted azobenzene compounds. The procedure described by Jaffe (59) was used to determine the slope,gj which has a value of 0.43 i 0.03. An attempt to examine the effects of steric influences on measured values of EJ/z and he was made by investigating 2,2'-dimethy1azobenzene (see Table VIII). Generally a given substituent operates with approximately equal strength from the ortho- and para- positions. 'Hence in the absence of steric effects one would expect that El/z and he for this compound would be nearly the same as for 4,4'-dimethy1- azobenzene. Experimentally, this appears to be the case, because El/z values for these two compounds differ by only about 70 mv, and kg for 2,2'-dimethylazobenzene is about the same as for 4,4'-dimethylazobenzene (0.13 and 0.15 cm sec- irespectively). Although sufficient data are not available, the implication of these results is that steric effects do not play a major role in determining the activa- tion energy for these electrode reactions. The data for 1,1'-azonaphthalene (see Table VIII) clearly cannot be used in above comparisons. However, it appears that a study of reactivity versus structure could be made with a series of substituted azonaphthalene compounds, because the mechanism for reduction is analogous to that of azobenzene. 97 Figure 17. Variation of kg with Hammett Q value for substituted azobenzene compounds (slope, 0.43 i 0.03; error level, i 10%). 98 —0.4 '— —O.5 - ém-O.6 '- 01 O H —O.7 P -0.8 _ I J l I -0.4 -0.2 0.0 0.4 99 A literature survey indicates that this is the first time linear free energy correlations with heterogeneous rate constants have been cited, and therefore these results are important for several reasons. For example, they imply that it may be possible to estimate values of kg for aromatic compounds whose Hammett sigma values are known, and kg for the parent compound has been measured. In addition, the results illustrate the type of structural changes that pro- duce large values of ks, something that is of considerable practical importance. Finally, they indicate that more detailed studies of the above type may provide valuable in- formation about the structure of activated complex during electron transfer, by, for example, comparing measured free energies of activation with quantum mechanical calculations for different configurations of the activated complex. MECHANISM OF REDUCTION OF AZOBENZENE IN WATER Reduction of azobenzene in protic solvents is a two electron process involving two hydrogen ions. Polarographic experiments have shown that the apparent reversibility of the electron transfer depends on hydrogen ion concentration, the electrode process becoming less reversible as pH is raised (23-33). To account for these facts some workers have suggested a stepwise reduction with a single hydrogen ion involved in each step (24), but no experimental data have been presented to prove or refute this idea. Thus, the possibility exists that in aqueous solvents the reduc- tion may involve two electrons with 0, 1, or 2 hydrogen ions in the rate determining step with the remaining hydrogen ions participating in preceding (and or succeeding equilibria. In an effort to resolve these possibilities cyclic voltammetry was used to measure apparent heterogeneous rate constants for reduction of azobenzene, and the dependence of these rate constants on pH was determined. Because available theory of cyclic voltammetry could not be applied directly to reactions involving hydrogen ion, it also was necessary to extend the theory to include these cases. Re- sults of this theory together with applications to kinetics of azobenzene reduction constitute the final part of this thesis. 100 THEORY Theory of cyclic voltammetry for the mechanism kf .__: R :11 O + ne is given by Nicholson (14). For this simple case theory takes the form of the following integral equation Y X(Y) (76$ H(Y)) «of? XiElQE._ Sl( (y) _ Y95x(y{§ .XLEIQE (32) w y-z 0 where ‘ =-—————1—-————- 33 X(Y) nFAC JFEBE' ( ) y 3 at (34) D 1/ 7 = (DR) 2 (35) w = ya ks/JwaDo (36) 79 = eXp (% )(Ei - El/z) (37) There g_is the transfer coefficient and ks the standard heterogeneous rate constant at §_= E1/2- The function SK(Y) is defined as eXp(-y) t < k = 38 Sk exp(-y - Zak) t > A ( ) where L_is the switching time. 101 102 Numerical solutions of Equation 32 are presented in the form of a working curve that relates peak potential separations, Agp, to the kinetic parameter, y_(details of this working curve already were discussed in the section dealing with measurement of ks values in DMF). The above theory clearly applies only to compounds that are reduced according to Mechanism II. For reduction of azobenzene the following more general mechanism must be considered K k K + 1 + 2 A + mH < > O + ne + pH < f> R + qH+':-—> Z .111 There g1 and .52 are equilibrium constants for possible preceding and succeeding reactions, 5f and.]_0+2e+H+-g—-£>R IV kb k + A+2e+H+R'+H R v kb k A + 2e + 2H+ f > R VI There A represents azobenzene, Q_is protonated azobenzene, Rf is the anion of hydrazobenzene, and R_is hydrazobenzene. To reduce further the number of possible mechanisms the value of g must be known. An approximate value Ofgl can be obtained from experimental current-voltage curves by a comparison with theoretical curves calculated for several values of g_and a given value of y, This compari- son is illustrated in Figure 21 for g_= 2.00 (determined from peak potential separation §_x agp = 72mv). The best agreement between theory and experiment is obtained for g equal about 0.7, which indicates that p_for the system is 2. Therefore, these data strongly suggest that both hydro- gen ions are involved directly with the electron transfer reactions, not in preceding or succeeding chemical equilibria --i.e. Mechanism VI above. The above data do not, however, distinguish between a concerted two-electron transfer and two stepwise one-electron additions each involving a single proton. Because a con— certed two-electron reduction is unlikely, the latter 115 Figure 21. Comparison of theory and experiment. 0, theory (g, 0.7; in 2.0) D, theory (g, 0.5; g, 2.0) A, theory (gj 0.3; g, 2.0) '———5 experiment for reduction of mg azobenzene _1 in 0.1M perchloric acid (scan rate, 106 mv sec ; Jail/2, 0.013 v _vg S.C.E.). uamp Current, 116 30 20 H O O l [.1 O -20 0 0 o O o O a O C) O C ' 0 0 o I I I I I I L 160 80 0 -80 (E - E1/2)n, mv Figure 21 117 mechanism seems most probable; in fact experiments in DMF to which various amounts of a proton donor have been added suggest stepwise reduction. Typical results of such ex- periments are shown in Figure 22. In Figure 22, curve A corresponds to reduction of azobenzene (given the symbol A in following mechanisms) in DMF with no added proton donor. These curves were discussed earlier where it was shown that Wave I results from formation of the monoanion of azobenzene _: A VII A + e and Wave II corresponds to formation of the dianion > A— VIII A + e < and Wave III is the reverse of Reaction VII. 0n addition of benzoic acid curve B of Figure 22 re- sults. Thus, the effect of proton donor is to increase the height of Wave I, with a corresponding decrease in Waves II and III. Also, a new wave, Wave IV, appears at more anodic potentials. These results can be explained by postulating that Wave I now results from the following parallel reactions A + e :-> A IX + A + e + H > AH X Because Ag_should be more easily reduced than A_(64), any A§_that is formed (the amount will depend on the ef- fective acid concentration) is reduced immediately + AH + e + H > AH2 XI 118 Hmum3Im©HEmEHomamaumEHUIz.Z .mEDHO> wn &om .UHUm UHONC Q MWOH .Q Uflom oaoucmn ZEOH .0 Uflom oaoucmnfimfla .m Uflom UHouch 0: .¢ .Aanuwm > ow .mumu :momv Uflom UHoucmn mo mucsofim mafimum> £DH3 mUHEmEHomamnumfiflplz.z cam mumuoano lumm anacoEEmamzummuuwu.mm.o :H mamucwnomm_flflv.o mo cofluoswmu Mom mfimumonmaom moouuomam xumcoaumum UAHomU .NN musmflm 119 O.NI «a madman uao> ..m.u.m;MMerHudmuom O.HI m.OI O.NI n.HI O.HI _ q _ _ _ >H HH HHH >H HHH OOH OOH OON O O H OOH dme n 'querxno 120 causing an increase in the height of Wave I. The decrease of Waves II and III follows directly because now less‘A- is present for reduction at Wave II, and oxidation at wave III. Also, Wave.IV is explained by assuming that it repre- sents oxidation of the 532 formed during Wave I by the re- verse of Reactions X and XI. On further addition of benzoic acid curve C of Figure 22 is obtained. This curve is consistent with the reactions already postulated where now there is sufficient proton con- centration that for Wave I the second of the parallel re— actions (Reaction X) predominates. Thus, Wave I results from Reactions X and XI, where Reactioan is rate determin- ing because of the difference of reduction potentials dis- cussed above for Reactions X and XI. Again,-Wave IV cor- responds to the reverse of Reactions X and XI. On addition of more benzoic acid no significant changes occur, probably because benzoic acid in DMF is a weak enough acid that further increases in its concentration do not increase proton concentration (65). These ideas are sub— stantiated by the fact that on addition of water curve D of Figure 22 results (also, curves essentially identical with A, B, C, and D of Figure 22 are obtained by using sulfuric acid as proton donor, in which case curve D re- sults without the addition of water). If this explanation is correct the postulated reactions readily explain the transition from curve C to curve D. 'Thus, as proton con- centration increases the rate of Reaction X increases, and 121 an anodic shift for Wave I results (see curve D). Because the rate of the reverse of Reactions X and XI is independent of hydrogen ion concentration, the potential of Wave IV does not shift until the whole process becomes reversible-—that is Agp between Waves I and.IV is 30 mv. Beyond this point the electrode reaction is perfectly reversible, both waves I and IV shifting 60 mv per unit change of pH, as already discussed (Figure 18). Although the above discussion is only qualitative, the proposed step-wise mechanism is consistent with two hydro- gen ions (p.= 2) involved in the electrode reaction. In addition, it appears to explain the transition from non— aqueous to aqueous experimental results. However, to prove this mechanism would require extensive additional theoretical calculations and experimental data. 10. 11. 12. 13. 14. 15. 16. 17. 18. LITERATURE CITED Kern, D. M. H., J. Am. Chem. Soc., 15, 2473 (1953); lg, 1011(1954). Konrad D. and A. A. Vlcek, Collection Czech. Chem. Commun., 28, 808 (1963). Kivalo, P., Acta Chem. Scand., 9, 221 (1955). Koutecky,£L, Collection Czech. Chem. Commun., 22, 116 (1955). Testa, A. C. and W. H. Reinmuth, Anal. Chem., 32, 1512 (1960). Dracka, 0., Collection Czech. Chem. Commun., 25, 338 (1960). Jaenicke, W. and.H. Hoffmann,-Z. Electrochem., 66, 803, 814 (1962). Herman, H. B. and A. J. Bard, Anal. Chem., 36, 510 (1964). Schwarz, W. M. and I. Shain, J. Phys. Chem., 69“ 30 (1965). 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Fragiacomo, Ric. Sci., 23x 139 (1952). Hillson, P. J. and P. P. Birnbaum, Trans. Faraday Soc., gfi, 478 (1952). Castor, C. R. and J. H. Saylor, J. Am. Chem. Soc., ‘15, 1427 (1953). Wawzonek, S. and J. D. Fredrickson, J. Am. Chem. Soc., 17, 3985, 3988 (1955). Nygard, B., Arkiv. Kemi, 29” 163 (1963). Markman, A. L. and E. V. Zinkova, J. Gen. Chem. U.S.S.R., 2_9.. 3058 (1959). Florence, T. M. and Y. J. Farrar, Australian J. Chem., 12.. 1085 (1964) . Holleck, L. and G. Holleck, Naturwissenschaften, 51” 212, 433 (1964). Holleck, L., A. M. Shams-Bl-Din, R. M. Saleh and G. Holleck, Z. Naturforsch., lgj 161 (1964). Holleck, L. and G. Holleck, Monatsh. Chem., 95, 990 (1964). Chuang, L., I. Fried and P. J. Elving, Anal. Chem., §1J 1528 (1965). -Banthorpe, D. V., E. D. Hughes and C. K. Ingold, J. Chem. Soc., 2864 (1964), and references therein. Oglesby, D. M., J. D. Johnson and C. N. Reilley, Anal. Chem., §§, 385 (1966). Carlin, R. B. and R. C. 0dioso, J. Am. Chem. Soc., lg, 2345 (1954). 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Chem., 61, 795 1965). Meek, D. W. "The Chemistry of Nonaqueous Solvents", J. J. Lagowski, Ed., Academic Press, New York, N.Y., 1966, p. 57. Delahay, P., "Double Layer and Electrode Kinetics", Interscience Publishers, New York, N. Y., 1965. Devanthan, M. A. V. and M. J. Fernando, Trans. Faraday Soc., 66, 368 (1962). Minc, S., J. Jastrzebska, and M. Brzostowska, J. Electrochem. Soc., 108, 1160 (1961). Frumkin, A. N., B. B. Damaskin, and N. Nikolaeva-Fedoro— vich, Doklady Akad. Nauk S.S.S.R., 115, 751 (1957). Grahame, D. C., Z. Electrochem., 66, 740 (1955). 57. 58. 59. 60. 61. 62. 63. 64. 65. 125 Hammett, L. P., Chem. Rev., 1_7, 125 (1935). Zuman, P., Collection Czech. Chem. Commun., 66, 3225 (1960). Jaffe, H. H., Chem. Rev., 66, 191 (1953). Holleck, L. and D. Becher, J. Electroanal. Chem., 6, 321 (1962). Newbold, B. T. and D. Tong, Canadian J. Chem., 66, 836 (1964). Hoogewerff, S. and W. A. vanDorp, Ber., 61, 1203 (1878). Nolting, E. and T. Stricker, Ber., 61, 3139 (1888). ‘Hoijtink, G. J., J. vanSchooten, E. deBoer, and W. Albersberg, Rec. Trav. Chim., 16, 355 (1954). Price, E., "The Chemistry of Nonaqueous Solvents," J. J. Lagowski, Ed., Academic Press, New York, N.Y., 1966, p. 92. APPENDIX A Numerical Solution of Equation 22 for the Potential Step Linear Scan Method The method of Huber (20) was used to evaluate numerically x(§) in Equation 22 of the text. The method approximates the function X(6) by a straight line of lepe g_on a given interval of width 6_along the E axis. Evaluation Of‘g defines the function XCEE) on each interval. Thus, X(66) on any interval, 6, is equal to‘6 times the sum of all previous‘g i x(at) = 5 Z k=1 Gk (1A) Equation 22 can be reduced to a recursion formula to calcu— late the individual values of g (20) k-1 k-1 1'00 '.§ Qi[Ak-i+1]'s(k5).§ ai[Bk—i+1] 1-1 1-1 “k = ' (2A) I5¢1 ' W1] + 5(k5) [591 ‘ 7i] where Aj : [éj] [CD]. - (Dj-l} + éCDj-l - [ll/3° — ¢j_1] (3A) Bj = [éj] {(Dj - ¢j_1}+ é¢j-1 - [ij — ¢j_1] (4A) j = k — i + 1 (5A) and - 1/2 ¢i = 2(10) (6A) 3 (i = (2/3)(ié) /2 (7A) 126 127 ¢i = v;/2 (k/a)-1/2 erf [(k/a)i<5]1/2 (8A) . 1/ ‘ / 1 m = 1530/5 3 2 erf[(k/a)ié] (2 -1/2‘1 eXp 'l-(k/a)i<5] (9A) y 5 at = 0 S(k0)= y B 6 0 < at : ax (1dA) V B 9 + 16 - ax at > a% where yB 2 exp (%%)(Ei ‘ E0) (11A) W39 = exp (12%)(Ei — E0 - ES) (12A) In the above equations "erf" represents the error function (16) andk/6_is the kinetic parameter (see text). APPENDIX B Computer Program Numerical calculations were performed on a Control Data 3600 digital computer. ’FORTRAN language was used in the program. To increase versatility of the program wave forms for cyclic voltammetry and potential step-linear scan technique were included. The appropriate wave form is selected through the CONTROL data card (0.0 for cyclic voltammetry and 1.0 for potential step-linear scan). GTLN defines the initial potential, THETA is the step potential, DEL is the width of the integration interval in 66_units, SQUIG is the number of equations solved to 2A.: SSCANS is the number of single scans, LIMIT is the total number of equations to be solved, and PSI is 5/6_(the kinetic parameter). NTOT is the number of runs for any or all different values of GTLN, THETA, DEL, SQUIG, SSCANS or LIMIT, and NRUN is the number of different kinetic parameters for a given set of the above variables. The output consists of statements of input parameters and five columns of calculated data that are labled CHI(N), CHI(N)—IT, (E-EO)*N, cz/c0*, and AT. These columns contain values of XCEE): X(66) for the ex— tension of the current—time curve, (676?) x 6, Cz/C: , and 66_respectively. 128 129 PROGRAM REVSUC DIMENSION SFN(1000),PH11(1000),PH12(1000),PSI1(1000), 1PSIZ(1000),Z(1000), CHI 1000),X(1000),SFIT(1000), 2X1(1000),CHIIT(1000),Zl 1000) . READ 407,IM,ID,IY 407 FORMAT(312) READ 404, NTOT 3 READ 404, NRUN 404 FORMAT (12) C GTLN IS INITIAL POT--THETA IS STEP POT c CONTROL = 0 FOR CYCLIC--CONTROL = 1 FOR STEP READ 405, GTLN, THETA, DEL, SQUIG, SSCANS, LIMIT 405 FORMAT(5F10.0,I10) READ 403, CONTROL 403 FORMAT (F10.0) D = DEL IF(CONTROL)21,21,22 21 CONTINUE 8:09 P = s + 1. Q-l. 1:1 31 SFN(I)=EXPF(S*DEL*SQUIG-DEL*Q) SFIT(I) = SFN(I) Q=Q+1. I=I+1 IF(Q-P*SQUIG)31,31,36 36 s=s+1. P = s + 1. IF(P-SSCANs)51,51,50 51 SFN(I)=EXPF(-P*DEL*SQUIG+DEL*Q) SFIT(I) = EXPF(-D*Q) Q=Q+1o I=I+1 IF(Q-P*SQUIG)51,51,61 61 s=s+1. P = s + 1. IF(P-SSCANS)31,31,50 22 CONTINUE I = 1 30 SFN I) = EXPF(-THETA) SFIT(I) = SFN I) I = 1+1 QPR = I IF(QPR - SQUIG)30,30,35 35 XYZ = LIMIT IF(XYZ - SQUIG)50,50,40 40 Q = SQUIG + 1. 45 SFN(I) = EXPF(-THETA + D*(Q - SQUIG)) SFIT(I) = EXPF(-THETA) Q = Q + 10 I = 1+1 IF(-THETA + D*(Q-SQUIG)) 45,45,50 50 C PSI 406 55 200 300 550 552 551 553 554 301 555 557 558 590 556 559 560 561 562 563 130 CONTINUE IS K/A READ 406 , PSI FORMAT(F10.0) E = PSI SQR = SQRTF(3,14159265/E) 0055 N = 1,LIMIT Q=N PHIl N) = 2.*SQRTF(Q*D) PSIl N) = (1./3.)*Q*D*PH11(N) PHIZ N) = SQR*PROB(SQRTF E*Q*D)) P812 N) = .5*(PH12 N) — EXPF(-E*Q*D))*PHIl(N))/E CONTINUE GT = EXPF(GTLN) A1 = D*PHIl(1) - PSIl(1) A2 = D*PH12 1 — P312 1) PRINT 200 FORMAT(1H1) PRINT 300 FORMAT(50X,21HREVERSIBLE SUCCEEDING//) IF(CONTROL)550,550,551 PRINT 552 FORMAT (55x, 11HCYCLIC SCAN///////////) GO TO 554 PRINT 553 FORMAT (563. 9HSTEP-SCAN///////////) PRINT 301,1M71D,IY FORMAT(100x,7HDATE 12,1H/12,1H/12////) 1F(CONTROL)555,555,556 SP = (GTLN - SQUIG*D)*25.68857 PRINT 557, GTLN FORMAT (1X, 10HLN(THETA)= F6.3/) PRINT 558, SP FORMAT (1x, 14HSWITCHING POT= F8.3, 12H Mv. PAST EO/) PRINT 590, SSCANS FORMAT (24H NUMBER OF SINGLE SCANs= F6.0///) GO TO 561 SP = 25.68857*(GTLN — THETA) DS = D*SQUIG PRINT 559, SP FORMAT (1x, 16HPOTENTIAL STEP = F8.3, 12H MV. PAST EO/) PRINT 560, DS FORMAT (1x, 22HDURATION OF POT-STEP = F8.3, 9H AT UNITS///) PRINT 562, DEL, SQUIG, LIMIT ,PSI FORMAT (1x, 7HDELTA = F8.4, 2x, 8HLAM8DA - F6.1, 2x, 17HLIMIT = 15,2x,5HK/A = F10.5////) PRINT 563 FORMAT (25x, 6HCHI(N), 7x, 9HCHI(N)-IT, 4x, 8H(E-ED)*N, 6x. 16HCZ/CO*, 10x, 2HAT///) DO 100 N= 1,LIMIT Z(N) = A1 + GT*SFN(N)*A2 Z1(N) = A1 + GT*SFIT(N)*A2 101 66 100 74 131 C2 N N 101 I N I (mason-no O 81 B1 + X(I)*(D*S (PHIl(M+1) 1(PSIl(M+1) - PSIl(M))) B2 82+x1(I) *(D*S*(PHIl(M+1) 1(PS11(M+1) - PSIl M))) C1 c1 + X(I *(D*S*(PH12(M+1) ) 1(P312(M+1) - PSIZ(M))) -c2 C2+x1(I) *(D*S*(PHI2(M+1 1(PSIz(M+1) - PSI2(M))) CONTINUE x(N) = (1,77245 - 81 — GT*SFN(N)*C1 )/z X1(N)= 1.77245 — 82 - GT*SFIT(N)*C2)/Z1 CHI(N) = D*X(N) + CHI(N+1) CHIIT(N) = D*X1(N) + CHIIT(N—l) POT = 25,68857*(GTLN+LOGF(SFN(N))) 81 81 +. x(N)*A1 c1 C1 +-x N)*A2 c1. 81 - C1 AT D*Q PRINT 66, CHI(N), CHIIT(N),POT,C1,AT FORMAT (20x, 2F14.8,F12,4, F14.8, F12.3) CONTINUE Nmm mwN-l IF (NRUN)2,2,1 NTOT = NTOT —1 IF(NTOT)74,74,3 CONTINUE PRINT 200 END FUNCTION PR08(A) C1 0.0705230784 c2 0.0422820123 c3 0.0092705272 0.0001520143 0.0002765672 0.0000430638 =1..— ))))))**(-16.) II II II II II II PHIl(M PHIl(M PH12(M PH12(M ))+ D*PHI1(M) - ))+ D*PH11(M) - )) + D*P812(M) — )) + D*PH12(M) — N) N) (1. + A* (c1 + A*(C2 + A* (CG + A* (C‘4 + A*(C5