TEuE-EQREFICAL EVALUATION OF EFFECTS QF AMALGAM FQRMATEON 0N STATEON‘ARY ELECTROEDE POLARQGRAPHY ‘e‘x’i'TH APFUCATEGN YO REDUCTION OF ALKAU METALS {N ACETQMFRELE The“: 5011' ”we Degree of M. S. MECHEGAN STATE UHWEIEETY Floyd Hilbert Beyerlein 1967 Ihnbm LIBRARY ' . Michigan State: University ABSTRACT THEORETICAL EVALUATION OF EFFECTS OF AMALGAM FORMATION ON STATIONARY ELECTRODE POLAROGRAPHY WITH APPLICATION TO REDUCTION OF ALKALI METALS IN ACETONITRILE by Floyd Hilbert Beyerlein The theory of stationary electrode polarography has been extended to include influence of amalgam formation. The mechanism treated is O+ne a: R(Hg) where the charge transfer is Nernstian and the electrode is spherical. Effects of finite electrode volume are shown to be negligible for reason- able experimental conditions, and therefore have been ignored. For the single scan experiment results are presented in terms of a semiempirical spherical correction term. For the cyclic experiment results are summarized in tabular form. Important results of the theory include the prediction that the ratio of anodic to cathodic peak currents is greater than unity. In addition, enhanced peak potential separations also are predicted under some conditions. The theoretical calculations have been tested experi- mentally for the reduction of cadmium at a hanging mercury drop electrode, and the agreement between theory and experi- ment is excellent. In addition, reduction of several of the Floyd Hilbert Beyerlein alkali metals in acetonitrile has been studied with stationary electrode polarography, and the theory of amalgam formation has been used to explain some apparent anomalies. THEORETICAL EVALUATION OF EFFECTS OF AMALGAM FORMATION ON STATIONARY ELECTRODE POLAROGRAPHY WITH APPLICATION TO REDUCTION OF ALKALI METALS IN ACETONITRILE BY Floyd Hilbert Beyerlein A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemistry 1967 VITA Name: Floyd Hilbert Beyerlein Born: April 15, 1942, in Frankenmuth, Michigan Academic Career: Frankenmuth High School Frankenmuth, Michigan--1956-1960 Michigan State University East Lansing, Michigan-~1960-1964 Michigan State University East Lansing, Michigan--1964-1967 Degree Held: B. S. Michigan State University (1964) ii ACKNOWLEDGEMENT . The author wishes to express his appreciation to Professor Richard S. Nicholson for his guidance and en- couragement throughout this study. Thanks are also given to Sandra M. Beyerlein, the author's wife, for her encouragement and understanding. iii TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . . . THE ORY O O O O O O O O O O O O O O O O O O O 0 O C 0 Boundary Value Problem. . . . . . . . . . . . . Integral Equation Form of Boundary Value Problem. . . . . . . . . . . . . . . . . . Numerical Solution of Integral Equations. . . . Results of Theoretical Calculations . . . . . . Single Scan Method . . . . . . . . . . . . Cyclic Triangular Wave Method. . . . . . . “PERIMENTALI O O O O O O O O O O O C O C O O . O O O Instrumentation . . . . . . . . . . . . . . . . Potentiostat . . . . . . . . . . . . . . . Function Generator . . . . . . . . . . . . Cell and Electrodes. . . . . . . . . . . . Chemicals. . . . . . . . . . . . . . . . . RESULTS AND DISCUSSION . . . . . . . . . . . . . . . Evaluation of the Instrument. . . . . . . . . . Conventional Polarography. . . . . . . . . Potentiostatic Electrolysis. . . . . . . . Stationary Electrode Polarography. . . . . Comparison with Experiment of the Theoretical Calculations for Amalgam Formation . . . . Electrochemistry of Alkali Metals in Aceto- nitrile. . . . . . . . . . . . . . . . . . Conventional Polarography. . . . . . . . . Stationary Electrode Polarography. . . . . Comparison of Stationary Electrode Polar- ography of Alkali Metals with Amalgam Formation Theory. . . . . . . . . . . CONCLUSION . . . . . . . . . . . . . . . . . . . . . LITERATURE CITED 0 O O O C O O O O O O O O O O O O O APPENDICES . . . . . . . . . . . . . . . . . . . . . iv Page 13 15 22 29 29 29 55 58 58 59 59 59 4O 4O 41 47 47 48 48 53 54 56 TABLE II. III. IV. LIST OF TABLES Page Empirical Spherical Correction Parameters as a Function of Potential. . . . . . . . . . . . . . 25 Peak Potentials and Ratio of Anodic to Cathodic Peak Currents as a Function of $0. . . . . . . . 25 Peak Potentials and Ratio of Anodic to Cathodic Peak Currents as a Function of 7 . . . . . . . . 26 Peak Potentials and Ratio of Anodic to Cathodic Peak Currents as a Function of EA' . . . . . . . 27 Variation of Triangular Wave Frequency with Time Constant of Integrator . . . . . . . . . . . . . 57 FIGURE LIST OF FIGURES Theoretical cyclic polarograms showing effects of $0 with 7 = 1. . . . . . . . . . . . . . . . Spherical correction as a function of potential Variation of spherical correction with $0 for v = 1 O O O O O O O O O O O O O O O O O O O O 0 Circuit diagram of potentiostat . . . . . . . . Circuit diagram of function generator . . . . . Stationary electrode polarogram for reduction of ferric oxalate . . . . . . . . . . . . . . . Stationary electrode polarogram for reduction Of cadnlium. O O O O O O O O O O O O O O O O O O Stationary electrode polarogram for reduction of sodium in acetonitrile . . . . . .’. . . . . vi Page 15 18 21 51 55 45 46 51 LI ST OF APPENDICES APPENDIX 4 Page A. Reduction of Boundary Value Problem to Integral Form. . . . . . . . . . . . . . . 57 B. Reduction of Boundary Value Problem to Two Simultaneous Integral Equations. . . . . . 65 C. Relation of the Functions (é! _ and 5U or r-ro (BE-r=ro to Current. . . . . . . . . . . . 65 D. Computer Program . . . . . . . . . . . . . 66 vii INTRODUCTION The original objective of this research was to extend the technique of stripping analysis (27) to determination of trace concentrations of alkali metals in nonaqueous solvents such as acetonitrile. Stripping analysis consists first of a constant potential concentration of the metal ion in a mercury microelectrode, such as the hanging mercury drop electrode (27). After the deposition step, the concentration of metal in the electrode is determined by stationary electrode polarography. In contrast with conventional polarography the influence of ohmic potential losses on stationary electrode polarog- raphy cannot be eliminated by simple application of Ohm's law (9,15,17). Therefore, with stationary electrode polarography it is essential that ohmic potential losses be negligible. In principle this could be a serious problem when working in nonaqueous solvents where low conductivities are encountered. However, by using a three electrode configuration, it is possible to compensate electronically for ohmic potential losses (2). Therefore, before studies of stripping analysis were begun an electronic instrument for recording stationary electrode polarograms was constructed. A description of this instrument together with its evaluation based on electronic tests and chemical experiments is described in a later sec- tion. Before attempting to analyze trace concentrations of alkali metals, conventional cyclic stationary electrode polarographic experiments with a hanging mercury drop electrode were performed on millimolar solutions of the metals in aceto- nitrile. Although in general the alkali metals were well- behaved, peak potential separations (difference of cathodic and anodic peak potentials) were considerably greater than the 57/n_mv. usually assumed for reversible electron transfer (19). This increased peak potential separation could not be accounted for by ohmic potential losses, because of the electronic compensation of ig_drop mentioned above. The ob— served peak potential separations also could not be explained in terms of kinetic effects of the electron transfer, because the peak potential separations were independent of scan rate (18). A third possibility was the fact that amalgam formation was taking place. All previous theoretical treatments of stationary electrode polarography--including the calculation of 57/g_mv. peak potential separations for reversible electron transfer—-have been based on plane electrode geometry, or in a few cases on spherical electrode geometry. In every case, however, effects of amalgam formation have been ignored. Ignoring amalgam formation for a plane electrode is justified because no mathematical distinction exists between the cases of Species soluble in the solution phase or electrode phase (6). Most applications of stationary electrode polarography, however, involve the use of spherical electrodes (hanging mercury drop electrode), and in this case a mathematical dif- ference does exist between the two caSes (28). In spite of this fact previous theoretical treatments of spherical electrodes have not considered amalgam formation (19), on the assumption that sphericity would be the only important effect. Recent work in other areas, however, indicates that this assumption may be in serious error. For example, Stevens and Shain (50) for the case of potentiostatic electrolysis and Delmastro and Smith (8) for the case of a.c. polarography have shown that in some cases consideration of amalgam forma- tion is essential to correct interpretation of experimental results. Therefore, effects of amalgam formation appeared to be a possible explanation for the observed behavior of the alkali metals. In addition to the effects described above, it recently has been shown that the kinetics of exchange reactions of metal ions with ligands such as EDTA can be studied by oxi— dizing the metal amalgams from a hanging mercury drop elec- trode with stationary electrode polarography (15,29). In each of these cases, however, theory was used which ignored effects of amalgam formation, again on the assumption that these effects would be unimportant. For these reasons it seemed important to investigate quantitatively the effects of amalgam formation for stationary electrode polarography with spherical electrodes before de- veloping stripping analysis methods. Therefore, the major portion of this thesis reports on the theory of stationary electrode polarography for amalgam formation, and as will be shown these effects can be very important to correct interpre- tation of polarographic curves. THEORY To treat rigorously the case of amalgam formation 0 + ne ‘——. R(Hg) I for stationary electrode polarography with a spherical elec- trode appears to be very difficult (8,25,50). However, following the discussion of Reinmuth (25) the problem can be simplified greatly by ignoring the influence of finite elec- trode volume and considering only the effects of sphericity and the convergent nature of the diffusion process. This restriction is perfectly justified for the case of stationary electrode polarography with a hanging mercury drop electrode, because Reinmuth has shown that for typical electrodes, effects of finite volume become important only for electroly- sis times of the order of 40 seconds. For stationary electrode polarography this corresponds roughly to scan rates of 8 mv./sec. or slower. Scan rates of this magnitude are at least a factor of three smaller than the slowest scan rates normal- ly employed. Therefore, the mathematical treatment which follows neglects finite electrode volume according to Reinmuth's suggestions. The only mechanism to be considered is I where the charge transfer is assumed to be Nernstian. Although the more general case could have been treated, the most pronounced effects of amalgam formation would be expected for the reversible case. Therefore, the treatment of the reversible case should serve to define qualitatively all of the trends to be expected for more complicated cases. Boundary Value Problem The boundary value problem based on Fick's diffusion equations for a spherical electrode, and considering the restrictions cited above, is §99 = BZCQ 2. 8C 8t D0[ or + r EEG] (1) 5CR _ 520R 2 5C (2) gr "DR[§FZ’+F 5:31 t = O; r 2_ro Co = 00* (5) t=07r20 CR-O (4:) t>o;r—>oo co—->co* (5) t>O;r—h0 cR—»o (5) . _ 5 ._ 5C (7) t>0,r-ro DOB-go-DR-a-ER 99.: BE. _ CR eXPI-(RTHE E0)] (8) In the above equations §_represents concentration as a function of time, t, and radial distance, r, from the center of a sphere of radius, £9. 90* is the initial analytical concentration of the oxidized species, 0, and it is assumed that the reduced substance, R, is generated in_situ (see Equation 4). The remaining terms and equations have their usual significance and are discussed by Reinmuth (24). Equation 8 is the Nernst equation consistent with our assumption of reversible electron transfer. For the case of stationary electrode polarography the potential in Equation 8 is a triangular wave function of time, that is E=Ei-vt o o; r = r0 a? = e Sx(t) (11) where e = exp [(%%)(Ei - E°)] (12) exp(-at) at g.aA Sk(t) = (13) exp(at-Zak) at 2_aA and a = fi¥l . (14) The above boundary value problem cannot be solved ana- lytically because of the nonlinear and discontinuous nature of Equation 11. Nevertheless useful numerical solutions can be obtained as described in the following sections. Integral Equation Form of Boundary Value Problem Although the preceding boundary value problem can be solved only by numerical methods of analysis, the numerical treatment is greatly simplified by first reducing the boundary value problem to integral equation form. Reduction to a single integral equation is developed in Appendix A. The result is (Equation A29): 1 t CO*7JB:=fO fO(T){J?Tt—?T—-— -g-exp[% 2(t-T)]erfC(J§E¢t-T ])dT ”SA—98% (t) t fo (T {WE-2:)— “ID—R eXP[LR2 (t- rHerfcfw—RJt-TJ + OZJDR r exp [- 232 (t-T)]] dT . (15) o 0 Because of the complex form of the kernels in integrals of Equation 15, direct numerical solution of Equation 15 would be very difficult, and therefore an alternate approach was sought. This alternate approach, which is described in Appendix B, resulted in a set of two simultaneous integral equations, each of which was considerably less complex than Equation 15. This system of equations is (Equations B4 and B5): av 68 (tN. DR ( —) ”ad'r A ft or r= = roC0* + 2Co*r56' Vt JV 0 Jt-Tr J}? in (16) -*JD‘ t ( r )r=rodT J-T: o Vt-T .1 _ _ _ (_D_Q _ DR ).es)\(t)~/DR r0 EE-r=ro r0 5r r=ro r02 rozesx(t) ~f__ v av ft (31—; r=erT 0 Jt—T (17) All of the kernels in Equations 16 and 17 are identical and considerably less complex than those of Equation 15. Therefore, Equations 16 and 17 are more amenable to numerical solution than Equation 15. Numerical Solution of Integral Equations For numerical solution of Equations 16 and 17, it is important to have these equations in dimensionless form so that results are not dependent on particular values of experi- mental parameters (A, Q, 2, etc.). Reduction to dimensionless form can be accomplished with the following change of variable T = z/a (18) and the following definitions y = at (19) _~/‘n$ 250 ' :07; (20) Do 7 =' i; (21) U _ 0 5m x (y) — comf? (5r,r=ro (22) w (y) = 919—— §Y—) (25) wow; 5r r=ro 10 Equations 16 and 17 now become (z)dz _ 2Jy U y X (zzdz GS (y) y 2_______ 1 + _____Q.- 24 ax (om T (om H and = [7¢0953x(y) 'go/PY] fy ‘i’(Z) dz Jfir- 0 y-z (25) 7x(y)-Y(y) respectively. The functions x(y) and Y(y), once calculated, are related to current by the following expressions which_are derived in Appendix C (Equation C2): i = nFACBJaDO F(y) (26) where _ _ _ _ 1(1) 95 1(y) F(y) —-J77'[X(y) (x(y) v ) (982%(y)-%2)] (27) Values of F(y) are directly related to potential by recalling the definition of Sak(y) (Equation 15) (E - E°)n = %1 ln esax(y). (28) Interestingly the functional dependence of current on experimental parameters can be deduced directly from Equation 26 without actually solving the integral equations. Thus, current is directly proportional to bulk concentration, Also, dependence of current on electrode geometry and hence amalgam formation is embodied in the dimensionless 1 (Equation 20). The magnitude of U in turn parameter, g; -0 11 depends on the experimental parameters £9 and 3% (see Equations 14 and 20). Thus, as either £9 or y_increases, .go approaches zero. With g0 zero, Equation 25 reduces to 7x(y) = Y(y). , (29) Equation 29 substituted into Equation 27 gives Fo(y) =IJET'x(y) (50) where the subscript on the function, Fo(y), is taken to mean the general current function, F(y), evaluated for‘gO equal zero. Combination of Equations 29 and 50 with Equation 24 results in the following single integral equation fY EQLELQE = 1 O “(y-Z 1+795a)\(y) (51) Thus, for the case of go calculated directly from Equation 51. Equation 51 is an Abel sufficiently small, currents can be integral equation, the closed form solution of which has been given previously (19). In addition, Equation 51 is exactly the equation which describes the case of a planar electrode of semi-infinite volume (19). Reduction of Equations 24 and 25 to this case is entirely reasonable, because go approaches zero as £9 approaches infinity in which case the electrode would become a plane of semi-infinite volume. For this case no distinction can be made between R soluble in the electrode or solution phase (6). 12 Results of these observations are twofold. First a check of the numerical solutions of Equation 24 and 25 is provided, because these results must reduce to previously published solutions of Equation 51 as g0 Also, the results indicate that the parameter approaches zero. g0 can be regarded as a spherical correction term which simultaneously includes effects of amalgam formation. Interestingly this parameter is of exactly the same form as the spherical term derived by Reinmuth (25) for the case of diffusion to a spherical electrode, but with both 0 and R soluble in the solution phase. The differences between these two cases is discussed in the following section. Aside from these limiting cases the exact form of the current-potential curves and their dependence on go can only be obtained through solution of Equations 24 and 25 explicitly for x(y) and F(y). Equations 24 and 25 were solved by two different numeri- cal techniques, the step functional method (19) and the method due to Huber (11). Solutions obtained with both methods converged to the same values, but because Huber's method is inherently more accurate, most of the results reported here were obtained by that method. All calculations were performed on the Michigan State University Control Data 5600 digital computer, and the FORTRAN program for Huber's method is listed in Appendix D. 15 Results of Theoretical Calculations Results of the numerical solution of Equations 24 and 25 in terms of the current function, F(y), for two values of g6 and 1_equal one are shown in Figure 1. The curve for Q6 = 0.001 is identical (within 1%) to the previously pub- lished solutions of Equation 51 discussed in the preceding section. In addition, for values of $6 3 0.001, the solution of Equations 24 and 25 was found to be independent of values of 1_and 9, provided 79 was larger than EEE (6.5). This ob- servation also is consistent with previous solutions of Equation 51 (25) and simply corresponds to the fact that polarograms are independent of initial potential provided the initial potential is sufficiently anodic. Further discussion of results of the numerical calcula- tions is most conveniently divided between the single scan and cyclic experiments. Single Scan Method. Effects of amalgam formation can be treated quantitatively only for the single scan method be- cause of the complicating influence of switching potential for the cyclic experiment. To determine the influence of amalgam formation on the single scan experiment, calculations were performed in which both E and 1_were varied independently. _O From variations of 1_with fixed gb effects of 7 were found to be relatively minor causing primarily small variations of the cathodic peak potential. For this reason and because 14 .mcmom vaponu 0 mo mcoamcmuxm mum mmcfla Umnmmn .H u % EDHB. 8 mo muommmm mcHBOSm mEmumoumaom UHHomo HMUHumHomSB .a musmflm 15 Oman a musmflm .>E .caflmlmv owl o¢l o 0* _ fl _ _ .O.o $.01 (KM N.o $.o 16 20 and ER usually are not markedly different, the remaining discussion is limited to the value of 1_equal one. In addi- tion only values of go in the range of 0E .GA&WLMV 00H: OMHI 00: 0m: 0m .N musmflm ‘\ llll“‘| 00.0 «0.0 , $0.0 00.0 00.0 0H.0 uorqoelloo Teorlequ 19 result is in sharp contrast to the case of amalgam formation where the spherical correction is negative for potentials anodic of E9, and is positive for potentials cathodic of E9. Moreover, the potential region over which the correction ranges from negative to positive is relatively small (ca. 50 mv.), which accounts for the distorted appearance of some of the polarograms (see Figure 1). Also, the magnitude of the spherical correction is actually larger near the peak than the maximum value of Reinmuth's correction, but approaches Reinmuth's value at negative potentials. Thus, the assump- tion often made that sphericity should have little effect on peak currents (20) is not valid when amalgam formation is present. Although the data of Figure 2 indicate the importance of amalgam formation, unfortunately such data are not very use- ful in practical terms because every value of go encountered experimentally would require a new computer solution of Equations 24 and 25. In an effort to circumvent this problem the dependence of the Spherical correction term on g6 at fixed potential was investigated, and typical results are shown in Figure 5. From Figure 5 it can be seen that the spherical correction is nearly a linear function of 0 , except for some potentials where deviation from linearity is observed for large values of go. Because these deviations from linearity are never large, however, it is possible to fit (within 1%) all of the data of Figure 5 to a parabola, and, thereby define empirically a spherical correction as 20 .2 small 3 .a n F Mom 00 QDHB coauomuuoo HMUHHmnmm mo COHDMHHm> .m musmflm 21 .m musmflm 0 0 00.0 00.0 8.0 00.0 00.0 «0.0 00.0 No .0 8.0 00.0 ) $.00 mw.mmd 00.0 mmé ||5N0.0 S d w II. «0.0 u. mm.m I m T.— o o 1 . II . m Amem mm mo 0 D .4 To 0 u I|,mo.0 ..|..o.n.o 22 spherical correction = a0: + 600. (52) There g_and §_are coefficients of the parabola and are func- tions of potential only. With the aid of Equation 52, there- fore, it is possible to define the current function, F(y), of Equation 26 as follows F(y) = Fo(y) + 003 + ago . (53) Since values of Fo(y) can be found in the literature, with the aid of Equation 55 it is a simple matter to calculate currents (see Equation 26) for amalgam formation provided the constants g_and Q are known. These values of g_and Q, accurate within 1%, are listed in Table I. For convenience the values of Fo(y) are also included. It should be emphasized that the data of Table I are strictly applicable only forjl equal one and g0.g 0.1, but as already mentioned this includes most cases of interest. Cyclic Triangular Wave Method. Data of Figure 1 show that amalgam formation is especially important for a cyclic experiment--that is, the anodic portion of the curve is even more strongly influenced than the cathodic portion. The reason for this is the product of electrolysis, R, is confined to the finite volume of the electrode. Thus, even though the diffusion process during oxidation is divergent, the actual anodic current is larger than it would be in the absence of amalgam formation, and the ratio of anodic to cathodic peak Table I. 1 __ 25 Function of Potentiala Empirical Spherical Correction Parameters as a (E-Eé)n, mv. Fo(y) a B 120 0.009 -0.004' -0.018 100 0.020 -0.005 -0.024 80 0.042 -0.011 -0.056 60 0.084 -0.055 -0.065 50 0.117 -0.068 -0.085 45 0.158 -0.095 -0.087 40 0.160 -0.122 -0.090 55 0.185 -0.162 -0.086 50 0.211 -0.211 -0.082 25 0.240 -0.270 -0.066 20 0.269 -0.540 -0.041 15 0.298 -0.424 -0.005 10 0.528 -0.507 0.042 5 0.555 -0.594 0.100 0 0.580 -0.676 0.174 -5 0.400 -0.749 0.261 -10 0.418 -0.805 0.548 ~15 0.452 -0.842 0.459 -20 0.441 -0.861 ‘0.520 -25 0.445 -0.862 0.615 -28.50 0.4465 -0.848 0.670 -50 0.446 -0.841 0.692 -55 0.445 -0.800 0.768 -40 0.458 -0.740 0.845 -50 0.421 -0.607 0.951 -60 0.599 -0.476 1.019 -80 0.555 -0.258 1.098 -100 0.512 -0.095 1.107 -120 0.280 -0.021 1.097 -150 0.245 -0.002 1.080 aCurrent for a spherical electrode is given by i . = nFACSJaDO [Fo(y) + a Do r0 a 7155 + B r07? ] 24 currents (anodic peak currents are measured to the extension of the cathodic curve (19)) can be greater than unity. Tables II, III, and IV summarize the effect of the three para— meters 0 and §_ (switching potential) on the ratio of —0’ 1’ A peak currents. The conclusion drawn from these data is that whenever Sphericity is important, the ratio of peak currents will be larger than unity. This fact is especially important because previous results indicated that only kinetic effects could cause the ratio to differ from unity (19). In fact, it was suggested that this ratio be used as a diagnostic test to demonstrate the presence or absence of coupled chemical reactions. In light of the present results, however, these diagnostic tests must be revised. The ratio of anodic to cathodic peak currents also has been used to measure homo- geneous rate constants (15,29). It now is clear that such measurements can be in error if amalgam formation is involved. Tables II, III, and IV also summarize effects of the three parameters 0 , y, and E_ on peak potential separations. These A effects on peak potentials all are consistent with the Nernstian model assumed for the electrode process. For ex- ample, for constant g0 and §_ (Table III) increases of the A parameter y_(VQO.WJQR) cause both peaks to shift anodically. This result is reasonable because an increase of 1_corres- ponds to a decrease of in which caSe the surface concentra— -D-R tion of R relative to 0 would increase (R diffuses into the electrode more slowly). This effect causes a Nernstian shift 25 .H u > was .>E m>.mmfil n cmflm I Kmv 00..“ 00.6.0 #013 mm.mmI 0.70 mw.d mm.am 04.4N mm.0mu mo.0 mm.a mm.am >m.mm mm.¢mI 00.0 Hm.a mm.am nm.mm mm.¢mu 40.0 oa.a mm.dm 4m.mm Ha.mmn No.0 mot—u 00.9.... 00.0w «umdml H00 oo.a mo.mm 0m.mm 0m.mml aoo.o m U Umfl\mmfl .>E .mma: .>E .caflm I mmv .>E .cmflm I mmv 00 m 00 mo Goauocdm m mm mucmuudu xmmm UHUOQDMU ou UHGOG¢ mo oHumm 0cm mamaucwuom Mmmm .HH magma 26 .00.0 n 00 000 .>a 00.005- u 3mm I <5 0 «m.a NN.¢0 am.mm am.mMI ¢.N mm.d mm.¢0 00.0N 0m.>ml N.N 0N.H NN.¢0 00.0w N¢.>ml 0.N 0N.H mm.¢m 00.0N 00.0ml m.a mm.d mm.dw «m.mm Hm.wMI ¢.H m¢.d mm.aw 04.0w mm.SMI 0.H m>.d mm.fim mm.am 0H.0¢I 0.0 mm.a mm.dm ww.md fiN.N¢I 0.0 >H.0d NN.¢0 00.0 >¢.mml N.0 omfl\mmfl .>E .mmda .>E .cawm Immmv .>E .camm Iommv k m 0 mo coauucsm 0 mm musmuuso Mmmm vaponumu ou UHUOQ¢ mo oaumm 0cm mamausmuom xmmm .HHH dance 27 U o .>E mm.smu u cams I so A .a u a cam 00.0 n 00 m 4m.a mm.a0 00.0w mm.0>HI m¢.H mm.H0 04.4w mm.0mHI m¢.d mm.dm 00.4w m>.mNHI mm.d mm.¢0 00.0N ma.aml mN.H mm.¢m hm.©m $0.00I Umfl\mmfl .>E .mmd a .>E .GAmH I mmmv .>E .cmflm I Kmv Q m Am mo coauocsm m mm musmuuso Mmmm Uflwonumu Op UHUocd Mo oaumm Ucm mamaucouom xmmm .>H mHQMB 28 of the waves along the potential axis to more positive values. Since both cathodic and anodic peaks shift simultaneously, the difference of the peak potentials remains relatively constant. Similar interpretations can be given for the in- fluence of 0 -0 and E A on peak potentials. EXPERIMENTAL Instrumentation As described in the Introduction electrochemical experi- ments performed in nonaqueous media require electronic compen— sation of ohmic potential losses. Because no suitable . instrument was available commercially, one was constructed from solid state Operational amplifiers (Nexus Model SA-1 except for amplifier §_of Figure 4 which was Philbrick Model P-2). Power for all of the operational amplifiers (i.15 volt, 400 ma.) was supplied by two Elcor zener-regulated power supplies (Elcor Electronics, Type A215-400). These power supplies were selected because of their excellent isolation characteristics from ground. The instrument which was constructed consisted of essen- tially two different sections, the three electrode potentio— stat and the function generator. For convenience each of these sections is discussed separately. Potentiostat. A block diagram of the potentiostat is shown in Figure 4. The circuit is of conventional design and its operation is described elsewhere (5). The load resistor, 3L, used to control current sensitivity was a Heath Model EUW-50 decade resistance box. 29 5O Hmcuoomu map on m umHMHHmEm muomacoo Axon wocmumflmmu opmuopv Houmflmmu Umoq HUAMAHQEM HUBOHHOM ucmnuso “mamaamfim Hm3oHH0m mmmuao> HUHMHHQEM Houucoo moonsom Hmcmflm HMQOADAUUM How manmaflm>m usmcfl muuxm Hmflucmuom HmHUHcH uSQCH Houmumamm coauoadm mponuomaw maflxHOB mpouuomam Hmuasoo wpouuomam mocmummmm "9mm "am "OMOU um u.¢.U ".mHm “Hm ".o.m "m3 "mo «mm .umumOHucmuom mo EMHmMHU HHSUHHU .4 musmflm 51 .w wusmflm [4h mmnmovmm m3 1) mm WU Re. mg. . UAW. .I<@<®mnv.on aa.zooa 4a.xooa J>))\|.0._m Ra.m00fi 52 The d.c. operating characteristics of the potentiostat were evaluated first with the aid of a resistive dummy cell. An accurately known d.c. potential was applied at the input labeled E5 of the control amplifier (C.A.) with a portable precision potentiometer (Biddle Gray Model CAT. 605014). The potential then was measured with a potentiometric re- corder (Sargent Model SR) at the output of the current follower (amplifier C.F.) for various values of 3L. These potentials always were consistent with the potential at Ei within the 1% tolerance of the resistors used. To ensure that frequency response of the potentiostat was adequate for the experiments to be performed (frequencies used never exceeded 25 Hz.) the rise time of the potentiostat also was measured. The high frequency characteristics of a potentiostat are determined by the bandwidth of the operational amplifiers and their output current capabilities (2). Because amplifier E had a relatively small bandwidth (75 KHz compared with 1.5 MHz for the others) and maximum current outputs were.i 2 ma., frequency response of the instrument was expected to be fairly limited. The rise time of the potentiostat was measured with a real cell which contained millimolar ferric ion in 0.2 M potassium oxalate-—0.2 M oxalic acid supporting electrolyte. The working electrode (WE) was a hanging mercury drop and the reference electrode (RE) was a saturated calomel electrode 55 with a Luggin capillary probe. All inputs in Figure 4 were grounded except the one labeled "SIG." at the input of amplifier C.A.; this input was connected to a pulse generator (Tektronix Type 161). The pulse generator provided a gate (rise time 0.5 microseconds) of 0.5 volt amplitude, which was sufficient to drive the potential of the working electrode well into the limiting current region for reduction of ferric oxalate. Rise time was then determined as the time required for the potential between the working and reference electrodes to reach 0.45 volts (90% of the potential applied at the in— put of the potentiostat). The rise time determined in this manner was 5 milliseconds, which corresponds to a useful bandwidth of about 55 Hz (2). Thus, the bandwidth of the instrument was suitable for all of the experiments described in this thesis. Function Generator. The circuit diagram for the function generator is shown in Figure 5. This function generator is similar to one described by Underkofler and Shain (51), and provides triangular and square waves in either a triggered or free—running mode. The circuit of Figure 5 actually differs in two major respects from the one of Underkofler and Shain. These differences are: (1) a ten turn potentiometer was in— serted in the integrator circuit (amplifier I) so that frequencies between various settings of the fixed input resistance and feedback capacitance could be obtained; and (2) the amplitude of the square and triangular waves was 54 H umamaamfim mom HOUHUmmmo Momnpmmw mo msam> muomamm H HmflmaamEm mo mocmumfimmu usmcfl mmmcmnu m>m3 Hmasmcmfluu Ho mumsqm.muomamm Auflsouflo CH new mUoE DOSm mamcflm mom .¢.Q HUHMHHQEM Op Hmmmfluu mmflamm¢ DHSUHHU may mo pdo Cam CH .mQ .mUOHU musm Hmmmfluu mo wuflumaom maouucou mUOHU mumum Uflaom mmUOHU Hmch HUHMHHQEM Houmscmuud Am>m3 Hoasmamauuv HmeHHmEm uoumumwucH Amm>03 mumsqmv HUHMAHQEM mocmHUMMHQ "mm "mm "mmm "mm “Dam "mam "mm "NQ.HQ "d " O¢OQ .Houmumcmm coauossm mo Emummap UHSUHHU .m musmflm 55 .m musmflm 12mpe zmav Moa mm BCBmOHBzmeom 0H. KOfi IJSSSS M000 Azmpe zmev Ma u xk.¢ s 0H Hum: fio. MAN 0 >0? Md ‘ .Illll£<<( 5 .0505 nvII2>2>II &H.MOH mm AnmmzHV 2a EVA 0 Mao >HQ mo AE 0.0m mumu :mom .HmucmEHummxm .mucflom .Hmoflumuomzu .mmcHA .mumamxo Ufluumm How EmumoumHom mUonuUmHm MHMGOHDMpm .0 musmflm 45 mm.0I 00.0I .0 musmflm muao> ..m.o.m m md.0I 05,0- 0 L aMN 0N.0I 0N.0I _ Q. s» _ __ H.0 N O 0.0 _¢.0 DVJu/T O * 009, 44 0.755 x 10-5 cm.2/sec. D ll 0 DR = 1.6 x 10’5 cm.2/sec. y = /I%Q-= 0.678. . R Because of the availability of these data reduction of cadmium under experimental conditions identical with those of‘Stevens and Shain was selected for quantitative comparison with theory. A number of current-voltage curves were recorded on a X-Y recorder (Honeywell Model 520), and a typical one is shown in Figure 7 (points). The experimental curve of Figure 7 was obtained at a scan rate of 21.5 mv./sec. with a hanging mercury drop electrode of radius 0.0676 cm. From these values of experimental parameters the value ofgO is calcu- lated to be 0.051. To compare the experimental curves of Figure 7 with theory, the current function, F(y), (see Equation 27) was calculated for 1_= 0.678 and = 0.051. With these values go of F(y), current was calculated from Equation 26 using the above values of electrode area, diffusion coefficient, and scan rate with the number of electrons, g, equal 2. The theoretical current calculated in this fashion has been in- cluded in Figure 7 (solid line). For Figure 7 the theoreti- cal polarogram was shifted arbitrarily along the potential axis to obtain the best agreement between theoretical and experimental peak potentials. This best fit corresponded to 45 .Hmucmaauwmxm .mucflom .Hmoflumuomnu .mmaflq .ESHEUMU mo coauoswmu MOM EmumoumHom mwouuomam humc0flumum .0 musmflm 46 .0 mnomflm 36> ..m.o.m .ml m m0.0I 00.0I 00.0I 00.0I fi0.0I m0.0I 00.0I 9 fl _ . ,. _ _ o 3 10.06.! 0 o I0.0I e 0 I00 6 .1. .D E c (10.0 e a e I0.0fi IO ,0 o [0.0a W. 0 I o 47 an apparent half wave potential of -0.645 volt s. S.C.E. compared with the literature value -0.642 volt s. S.C.E. (10). The excellent agreement between theory and experiment shown in Figure 7 establishes the validity of the theoretical calculations. This agreement between theory and experiment should be indicative in general of the extent to which theo- retical calculations for stationary electrode polarography can be expected to apply to real systems, provided the cor- rect theoretical model has been chosen. Electrochemistry of Alkali Metals in Acetonitrile Acetonitrile was used as solvent for study of three of the alkali metals (sodium, potassium, and lithium). Aceto- nitrile was selected because it has been shown to be a useful solvent for electrochemical studies, and moSt of the alkali metals already have been studied polarographically in acetoe nitrile (4). The alkali metal solutions were prepared from the anhy- drous perchlorate salts. A discussion of the electrochemical investigations of the alkali metals presented according to the experiments performed follows. Conventional Polarography. Polarographic investigation of the alkali metals was used to confirm previous work which indicated that alkali metals give well-defined polarographic 48 waves in acetonitrile (4). Polarograms were obtained for millimolar solutions, and results were in good agreement with the literature. Stationary Electrode Polarography. The analytical ap— plications of stationary electrode polarography are similar to those of conventional polarography where peak current is the parameter analogous to limiting current. For example, from Equation 26 peak current should be a linear function of bulk concentration in the case of reversible amalgam forma- tion. Thus, if reduction of the alkali metals follows this model, peak currents for the alkali metals should provide a means of quantitative analysis of the metals. Actually, from the analytical viewpoint stationary elec- trode polarography has at least two advantages over conven- tional polarography. First, the analysis time is significantly less; and second, under optimum conditions the technique is actually more sensitive than conventional polarography (7). For these reasons the dependence of peak current for the alkali metals was investigated over a range of concentrations (1.0 x 10'3 M.to 5.0 x 10'5 M). Within this range peak cur- rents varied linearly with bulk concentration for each of the metals. Thus, stationary electrode polarography can be re- garded as a useful analytical technique for alkali metals in nonaqueous media. Comparison of Stationary Electrode Polarography of Alkali Metals with Amalgam Formation Theory. The alkali metals are 49 known to behave reversibly in acetonitrile (12), and since they form amalgams readily we also compared theory with ex- periment for these metals. Values of 20 were calculated from the cathodic peak cur- rent of a stationary electrode polarOgram and Equations 26 and 51. Values of 2R were obtained from material transport measurements in liquid amalgams (26). As an example, typical of the other alkali metals, the results for sodium will be used for purposes of illustration. For sodium the following values of 20 and D were found -R D0 = 0.58 x 10"5 cm.2/sec. DR = 0.86 x 10“5 cm.2/sec. giving 7 = E? = 0.668. A comparison of theory with experiment is shown in Figure 8 for a millimolar solution of sodium with 0.1 M tetraethylammonium perchlorate as supporting electrolyte. The solid line is theoretical and points are the experimental polarogram. The experimental curve of Figure 8 was obtained at a scan rate of 161.4 mv./sec. with a hanging mercury drOp electrode of radius 0.0552 cm. From these values of experi- mental parameters 0 was calculated to be 0.007. _0 Calculation of the theoretical curve was the same as already described for the case of cadmium, except the above 50 .Hmucmfiflummxw .mucflom .Hmoflumuomnu .mmcHA .mafluuflcoumom CH ESHUom mo aoauosnmu How EmumoumHom mwouuomam 0HMGOHumum .0 musmflm 51 o m QHSEHM muHo> 70.0.0 .WN. m oo.NI mm.dI om.aI 00.6I 00.AI 00.HI O O O ' III1,O.OHI (1 O I; v, . 0 10.0] o I o .l..o.o o H... HI. 9 O 10.0 e O O O I|10.0.W O .o O J0.0H 52 values of go and y_were used. The theoretical curve was shifted along the potential axis to obtain the best agreement between theoretical and experimental peak potentials. This best fit corresponded to an apparent half wave potential of -1.878 volt yg, aqueous S.C.E. compared to the literature value of -1.855 volt yg, aqueous S.C.E. (4). Although the agreement between theory and experiment shown in Figure 8 is satisfactory, it clearly is not as quanti- tative as in the case of cadmium reduction (Figure 7). In particular, the peak potential separations are slightly larger than theory would predict. However, in view of the way in which diffusion coefficients were obtained for this system, these minor differences are not unreasonable. CONCLUSION Based on theoretical and experimental results presented in this thesis the importance of amalgam formation for stationary electrode polarography with spherical electrodes has been established. The most important effects are on ratios of anodic to cathodic peak currents and peak potentials. In both cases consideration of amalgam formation has been shown to be essential to correct interpretation of experi— mental results. When these effects are considered the agree- ment between the theory presented and experiment is excellent. The theory developed also is capable of explaining some appar- ent anomalies associated with reduction of alkali metals in acetonitrile. 55 1. 2. 10. 11. 12. 15. 14. 15. 16. LITERATURE CITED Alberts, G. S., and Shain, I., Anal. Chem. 55, 1859 (1965). Booman, G. L., and Holbrook, W. B., Anal. Chem. 55, 1795 (1965). Churchill, R. V., "Operational Mathematics," p. 55, McGraw-Hill Book Co., New York, 1958. Coetzee, J. F., McGuire, D. K., and Hedrick, J. L., J. Phys. Chem. §1J 1814 (1965). DeFord, D. D., Division of Analytical Chemistry, 155rd Meeting, ACS, San Francisco, Calif., April.1958. Delahay, P., "New Instrumental Methods in Electrochemistry," p. 52, Interscience, New York, 1954. Ibid., p. 140. Delmastro, J. R., and Smith, D. E., Anal. Chem. 58, 169 (1966). DeVries, W., and Van Dalen, B., J. Electroanal. Chem. 19, 185 (1965). Frischmann, J., Ph. D. thesis, Michigan State University, East Lansing, Mich., 1966. Huber, A., Monatsh. Mathematik und Physik. 41, 240 (1959). Kolthoff, I. M., and Coetzee, J. F., J. Am. Chem. Soc. 12. 870 (1957). Kuempel, J. R., and Schapp, W. B., 155rd Meeting, ACS, Miami, Fla., April 1967. Meites, L., "Polarographic Techniques," p. 56, Interscience, New York, 1955. Ibid., p. 71. Ibid., pp. 250-295. 54 17. 18. 19. 20. 21. 22. 25. 24. 25. 26. 27. 28. 29. 50. 51. 55 Nicholson, R. 8., Anal. Chem. 51, 667 (1965). .;5;5,, p. 1551. Nicholson, R. S., and Shain, I., Anal. Chem. 55, 706 (1964). Nicholson, R. S., and Shain, I., 5555,, 51, 190 (1965). O'Donnell, J. F., Ayres, J. T., and Mann, c. K., Anal. Chem. fl. 1161 (1965). Olmstead, M. L., and Nicholson, R. 8., Anal. Chem. 55, 150 (1966). Reinmuth, W. H., Anal. Chem. 55, 185 (1961). Reinmuth, W. H., £555., 54, 1446 (1962). Reinmuth, W. H., J. Am. Chem. Soc. 12, 6558 (1957). Schwarz, W., Z. Elektrochem. 55, 555 (1955). Shain, I., in "Treatise on Analytical Chemistry,“ Kolthoff and Elving, eds., Part I, Sec. D—2, Chap. 50, Interscience, New York, 1965. Shain, I., and Martin, K. J., J. Phys. Chem..§§. 254 (1951). Shuman, M. S., Shain, I., Great Lakes Regional Meeting, ACS, Chicago, 111., June 1966. Stevens, W., and Shain, I., Anal. Chem. 55, 865 (1966). Underkofler, W. L., and Shain, I., Anal. Chem. 55, 1778 (1965). APPENDICES 56 Reduction of Boundary Value Problem APPENDIX A to Integral Form It is convenient first to reduce Equations 1 and 2 of the text to parabo accomplished with U(r,t) V(r,t) lic form. This transformation can be the following functions rCO(r,t) rCR(r,t). (A1) (A2) In terms of the functions g_and y, the boundary value problem ‘ given by Equations 1 through 8 of the text becomes an _ 020 (St — D0(6r2) av = 0 v (5;) DR(5;29 t = 0; r > r0 U = rCE t = o; r 2_o v = o t > o; r -$-oo U -’-rCS t > O; r “9'0 V —”0 Do 0U Do t > o; r — r0 r0 (5r)r=ro - r02 Ur_r0 . DR 82. _ DR ;;'(6r)r=ro ”—5 Vr=ro 57 (A5) (A4) (A5) (A6) (A7) (A8) (A9) t > o; r = r0 —£5£Q = esk(t) .(A10) Equations A5 and A4 can be integrated easily with the aid of the Laplace transformation, for which we adOpt the following definition and notation _. co ;E[U(r,t)} = U(r,S) = U = f [exp(-St)][U(r,t)] dt 0 -(A11) Thus, taking the Laplace transformation of Equation A5, and applying Equation A5 one obtains 625 SUI_ £_ * 32' - DO - DO CO .(A12) A general solution for Equation A12 is * C r U'= -g—-+ A exp(r dS/DO) + B exp(-r-JS/DO) (A15) where §_and g are integration constants. From Equation A7, however, A_is evidently zero, so that Equation A15 becomes __ Car U =-§- + B exp(-r 757D; ) .(A14) The value of §_can be determined by evaluating Equation A14 at _F_Q * B = (U. - roCo ) exp (r0 JS/DO ) .(A15) S Equations A14 and A15 combine to give -)(- Cor S * E = + (UFrO - Inga) exp[ ./‘_‘s/DO (rO-r)] .(A16) Because current is calculated in terms of flux at the 59 electrode surface, Equation A16 is differentiated with respect to g-and evaluated at gfigo.‘ The result is *- BU' = C0 - rQCE (EF'r=ro 5—-— 73/50 (Ur=ro - s ) .(A17) With the aid of Equation A1, Equation A17 can be written in terms of concentration and flux of substance 0 EEO =cgvb'; [ 1 * 1 We "—7, s(“/D—_O+~/§_) +Co ENE-“7%) r0 3 _ 1 f0 (5) (A18) 755 Js_+~/—700'ro where the function fo(£) is the surface flux of 0: = 6CQ(r,t) fo(t) Do [ 5r ]r=ro .(A19) The inversion of Equation A18 to the real time domain can be accomplished with the aid of tables of Laplace transform pairs and the convolution theorem (5) t _ * _ 1 COr=r _ CO D f0 fO(T) J‘—' Do .r—' [:-——3¥———-- —QQ- exp(-—2 (t-T»erfc( EQJJt-T )] dT W r0 1‘0 ro Treatment of the equation in V(£,§) is similar to that for U(£,§). Application of Laplace transformation to Equa- tion A4, together with Equation A6, leads to the equation analogous with Equation A15 60 Vl= C exp(-r JS/DR )+'D exp(rnJS/DR) .(A21) Application of boundary condition A8 gives the following relationship between integration constants The remaining integration constant can be determined as before, and the final result is exp(-rIJS/DR ) - exp(r'JS/DR ) exp(-ro «ls/DR ) - exp(ro 'JS/DR )