Electrochemical methods available for reaction mechanism studies

The purpose of this paper is to review for chemists who are not specialists in electrochemistry the electrochemical methods presently available for st...
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ELECTROCHEMICAL METHODS AVAILABLE FOR REACTION MECHANISM STUDIES' A. EDWARD REMICK Wayne State University, Detroit, Michigan

Tm purpose of this paper is to review for chemists who are not specialists in electrochemistry the electrochemical methods which are presently available for studying reaction mechanism. Electron-transfer reactions, both organic and inorganic, will naturally occupy the center of the stage, but electrochemical methods are not necessarily restricted to these; neither are they necessarily restricted to heterogeneous reactions occurring a t the surface of an electrode. It will not be the purpose of this paper to review all of the reactions whose mechanisms have been studied with the aid of electrochemical techniques although a few such reactions will be used as examples. The goal of all mechanism studies is to establish the stoichiometric equation for each component step in an over-all reaction, to establish the order of the steps, to measure the forward and (if any) backward rate constants for each mertsurably slow step, and to measure the free energy change (or, alternatively, the reduction potential, in case an electron-transfer step is being considered) for a11 steps which are rapid enough in both directions so that they achieve essentially a state of equilibrium. It should perhaps be emphasized that to write a stoichiometric equation for any step requires the formula (preferably the stmctural formula) for each participating species, including a knowledge of its charge type and its state of solvation or other type of complex formation. Furthermore, electrons are conventionally considered as participating species in certain kinds of electron-transfer steps, and it is therefore important to establish the number of electrons involved in each elementary act of electron transfer. POTENTIOMETRIC METHODS

Reversible Reactions. The potentiometric method is the classical method of measuring reduction potentials and is so well known and has been so frequently reviewed that there is no need here to dwell on it a t length. It is the method par excellence for studying free energy relations in reversible redox systems. Strictly speaking, of course, classical thermodynamics cannot establish the path taken by a reaction since thermodynamic properties are independent of the path. Nevertheless, the fulfillment of specific thermodynamic rdationshios often eives us soecific information of value

' Based on 8. paper presented as part of the Symposium on Contributions of Spectroscopy and Thermodynamics to the Teaching of Organic Chemistry a t the 128th Meeting of the American Chemical Society, Minneapolis, September, 1955.

in the complete characterization of a given mechanism. For example, reversibility is established by demonstrating the applicability of the Nernst equation or other thermodynamic equations such as the Peters equation (1). The latter is used especially for organic reactions in which hydrogen ions are involved. Expressed in terms of the European conventionZrelating to sign, it is:

in which EEis the reversible reduction potential relative to the standard hydrogen electrode, and S represents the stoichiometric concentration. The last term is a function of the actual concentration of hydrogen ions and the dissociation constants. Its precise algebraic form varies with the type of oxidant and/or reductant involved ( 2 ) . It is obvious that EO, the standard electrode potential, and n, the number of electrons transferred per mole of the participating species, can readily be determined from measured values of Ex and known values of the redox ratio, S,/Srea, if measurements are made at constant pH. The values of pK, and pKz can be identified with the pH values a t the points of intersection of the extrapolated linear portions of the curves obtained when EO is plotted against pH. A quite different example of the use of potentiometrically determined reduction potentials in obtaining information of mechanistic interest is the wellknown series of investigations carried out by Michaelis and his collaborators (S), and independently by Elema (4), on the reversible two-step oxidations of pyocyanine and other substances which were shown t o yield semiquinones as intermediates. They found that the stability of semiquinones is a function of pH. By choosing a p H value corresponding t o as great a stability as experimentally practicable, they were able to show that these oxidations occurred in two distinct steps of one electron each which were compatible with thermodynamic equations. Here, as in d l cases involving equilibrium measurements, the validity of interpretations postulating the existence of certain ionic or molecular species rests upon concordance of experimental results with thermodynamic equations of one or another distinauishina mathematical form.

'The European convention lists standard potentials for strongly oxidizing systems as positive and for strongly reducing systems as negative. The American system is the reverse and signs in equation (1). would require reversal of 564

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VOLUME 33, NO. 11, NOVEMBER, 1956

Whether or not these particular species are the "participating species" in the electron-trausfer step cannot be determined by thermodynamics alone, but it is certain that these relatively stable species are a t any rate involved in what might he called "participating equilibria." As such, they contribute to the over-all picture, both thermodynamically and kinetically, and hence further our efforts to attain what we have called "the goal of all mechanism studies." A closely related example of interest is the dimolecnlar reduction of thioindigodisulfonate by hydrogen catalyzed by platiniaed asbestos (5). The thermodynamic equation: Ea

= Eo'

- 0.0245log (Q'zl -') - 0.0295log n

(2)

shows that in this case, in contradistinction to the examples already discussed, the reduction potential is a function of the dilution. Eofis the standard potential a t constant pH, a' is the moles of dye reduced by the addition of x of oxidant, and u is the volume of solution. As a further contribution t o the general problem of determining the equilibrium concentrations of ionic species present in solution, it should be emphwized that the potentiometric method is practically the only method available for examining dissociation equilibria involving complex ions. For example, the complex ion Ag(CN)%- dissociates to produce a certain concentration of silver ions which cannot be measured by a conductance method because of the relatively high concentrations of other ions present. It can, however, be measured potentiometrically by using a silver electrode which behaves reversibly relative to silver ions alone (6). Irrevemz72e Reactions. The potentiometric method was used by Conant and his collaborators (7) to study the rates of oxidation or reduction of organic compounds. Platinum electrodes were immersed in a poised solution (containing equal concentrations of the oxidized and reduced components of a reversible redox system) which was connected with a reference electrode. The organic material was then dropped into the poised solution and the rate of change of potential followed potentiometrically. The same organic substance could be oxidized (or reduced) by a whole series of poised solutions, thereby establishing a relation between reduction potential and the over-all rate of a homogeneous redox reaction (not a heterogeneous one occurring a t the electrode's surface, presumably). Conant argued that such a relation should be expected in case the reaction mechanism was of the following type:

haved in accordance with the mathematical requirements of equation @),the mechanism must be of the above type. He predicted that if a case were found in which a fast electron-transfer step is preceded by a slow step not involving electron transfer there would be no relation between potential and over-all rate. His prediction was fulfilled eventually by Weissherger (8) who showed that the rate of oxidation of benzoin is independent of the potentials of the oxidizing agents used and that the rate-controlling step is an enolization. THE THEORY OF ABSOLUTE REACTION RATES

Although this theory (9) is common knowledge as it relates to homogeneous reactions, a brief explanation of its important role in the development of theories of electrode kinetics will be given here for the nninitiated. The potential-energy diagram for the process of discharging an ion at an electrode involves two humps, one for the transfer of the ion to the electrode surface accompanied presumably by desolvation, the other for the electron-transfer step. The ordinate of this diagram is, as always, the potential energy; the abscissa may be taken as the distance from the electrode. A potential difference, E, exists between the electrode and the ions in the bnlk of the solution. This potential difference, depending on its sign, can help or hinder the surmounting of the energy humps. I n view of the fact that an ion starting in the bnlk of the solution (at the minimum of the curve) will traverse only part of the distance t o the electrode in climbing the humps, only a fraction (w) of the potential difference, E, helps the activation process. The remainder (1- a ) E hinders the reverse process which involves the same activated complex. When this effect of the potential is superimposed on the ordinary absolute-reaction-rate equations as they apply in the absence of the potential. there are obtained equations for the forward and reverse rates of reaction. Since current is proportional to rate, these equations are readily transformed into the following ones (for the cathode): + i

= nSAC,,kfo exp (unSE/RT)

C

i

=

- o)nSE/RT]

nSAC..a kbO exp [-(I

(4)

(5)

where the total current, i, is the algebraic snm of its --+

+

+

two components, i.e., i = i i (C's are actual concentrations, k's are chemical rate constants). By +

convention the reverse current, i, is negative. Equations (4) and (5) constitute the entering wedge enabling us t o relate electrical quantities t o chemical rate Ox, + Red* Red, + Oxp (fast) ( 3 4 constants. They are the unique contribntion of the theory of absolute reaction rates to electrochemistry. Red, Y,etc. (slow) (35) At equilibrium, the net current is zero and this because the rate is proportional to the concentration equation becomes : of the intermediate species, Redl, and this in turn s=-z= io (6) depends on initial concentrations and the potential This equation defines the exchange current, io. of the reversible O x ~ R e d system. , Conant seemed to The exact meaning of the heterogeneous rate contake the attitude that if an organic compound be-

-=

--

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stants, klo and kno,is made clearer by the following equations. k,

= klo exp

en3E (%)

(forward reaction)

k~ = kto exp [(I - G ) ~ ~ ~(reverse ] reaction)

(6a)

(6b)

kl and kB are functions of the potential whereas klo and kaOare not. The fraction, ru, is known as the "transfer coefficient." It may be determined from the Tafel equation written in the form (41): RT q = constant - - log i aF

(7)

I t is evident that a plot of the overvoltage, q, against the log of i, the net instantaneous current, gives a straight line from the slope of which ru can be determined. VOLTAMMETRY WITH DIRECT CURRENT

Introductia. Voltammetry is that branch of electrochemistry dealing with the relations between voltage and current in electrolytic cells. The subject has been reviewed recently by Delahay in his excellent book (10). Polarization of an electrode during electrolysis is well known to be caused by some slow step in the overall process of electrolysis. The rate-coutrolling step might be diffusion and always is a t sufficiently high current densities, or it might be electron transfer or some other slow step a t the electrode surface. If electron transfer is the slow step, the theory can be formulated mathematically in terms of the theory of absolute reaction rates. If diffusion is rate controlling, Fick's laws of diffusion are used as a starting point in the mathematical development. Fick's first law involves the flux, q, which is the number of moles of diffusing substance crossing one square centimeter of an imaginary plane per second. The law states that the flux is proportional to the concentration gradient. The constant of proportionality, D, is called the diffusion coefficient.

stirring results in a steady state in all cases. The dropping mercury electrode, as used in polarography, neatly avoids the problem of the steady state by yielding the average current over a recurrent time interval, the "drop time"; or, with suitable instrumentation, it can be made to yield results at some definite instant of time measured from the start of the drop formation. Thus, instead of giving results in terms of a steady state it gives them in terms of a reproducible state (11). Polarography and Reversible Reactions. The polarograph serves as a convenient although less accurate substitute for the potentiometer in the study of reversible redox systems. The number of electrons (n) transferred may be obtained by use of the Ilkovic equation, one form of which is (10): i

=

7.082 X 10' nmVz t X / a D'/=Co

(8)

where m is the rate of flow of the mercury (g. sec. -I), t is time, Co (mol. c m . 3 is the concentration in bulk of solution, and D is the diffusion coefficient. The polarographic criterion for reversibility is obedience to the equation: RT i E = El/, - - ln -;-n5

z n - i

(9)

where idis the diffusion current, coupled with the requirement that the half-wave potential, El/,, he the same for a given reduction and the reverse oxidation. The value of n is readily determined from the slope of the line obtained by plotting E against the logarithmic term of equation (9). The half-wave potential is related to the standard potential of Nernst which involves the activity coefficients j and corresponding diffusion coefficients D by the equation:

Some reactions satisfy equation (9) but are not reversible from the potentiometric viewpoint. The reason for this discrepancy will become apparent from examples to be given later. I n such cases, equation (10) does not apply, which is t o say that the half-wave potential may not be as close to the standard potential, EO, as it would be for a reversible reaction if the acNote that both q and C are fuuctious of time (t) and tivity coefficients and the diiusion coefficients had distance (x) from the electrode surface. The second nearly the same values for oxidant and rednctant. law shows how the concentration changes with the When equation (10) does apply, it is clear that if pH alters the half-wave potential, hydrogen ions must be time: involved in the reaction (cf., also, equation (1)). Irreuersible Reactions. Reversible reactions, as previously remarked, are not nearly as important as irreThe solution of the differential equation representing versible reactions from the standpoint of mechanistic Fick's second law yields, by using appropriate boundary studies. Electrochemical redox reactions involving one conditions, an equation involving the error function. or more slow steps are amenable t o voltammetric study Equations of this type, solved for electrodes of dif- and it will be of interest to consider the main methods ferent shapes, show that a steady state cannot be used in studying them and t o seek such diagnostic achieved in quiet solutions if plane or cylindrical criteria for differentiating mechanisms as these studies electrodes are used. Only with spherical electrodes afford. Obviously there must he possible a very large numwill a steady state be attained. However, rapid

VOLUME 33, NO. 11, NOVEMBER, 1956

ber of varieties of mechanisms involving electrochemical electron transfer and one or more slow steps. The most studied classes are the following, in which charge types are not indicated. They will be written as reductions, but corresponding oxidations are, of course, possible.

a slow, bidirectional electron transfer is the only step. Polarographers call such a reaction "irreversible"; if kl>>ka,they call it "totally irreversible."

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alternating currents) available for ascertaining whether the potential-determining step is controlled by diffusion or by activatiou (in the Arrhenius sense). I t is well known that the total overvoltage, 7, is made up of three components depending respectively on ohmic losses at the electrode surface, on concentration polarization, and on activation polarization. The relationship may be expressed by the equation: (15) n = n a n. .n

+ +

These three types of overvoltage can best be distinguished by their discharge characteristics, i.e., by the manner in which the voltage decays after cessation k, of the polarizing current (12). The ohmic overvoltage Class B Y F? Ox (dowp) (Iza) (7.J decays instantaneously, the concentration overka Ox + n e -Red (fast) (12b) voltage (?,) decays slowly and in a complex way, whereas the activatiou overvoltage (?.) decays exponentially. The participating species in the reduction is furnished Another method utilizes the temperature coefficient of by a slow "preceeding reaction." The precursor sub- voltage, dE/dT, which is several times larger for an stance, Y, in the kinetic process is not reducible a t the activation-controlled discharge than it is for a reversible electrode under the prevailing experimental conditions. process (13). Other ways in which concentration Polarographers refer to this class of reaction as a polarization can he recognized involve the reduction in "kinetic process" and to the current produced thereby overvoltage caused by rapid stirring or by irradiation as a "kinetic current." with ultrasonic waves (14). Finally, polarographers Class C Ox, + ne Red, (fast) (13a) have shown that the current is proportional to the square root of the head of mercury used in the capillary k/ tube of the dropping electrode if the process is diffusionRed, + Oxt e Ox, Red* (slow) (13b) controlled but is independent of the head if activaks tion control obtains (15). The characteristic feature of reactious of this class is ~ ~ t now~ to voltammetric ~ n i methods, ~ ~ it should the partial regeneration of the initial oxidant as the he remarked that they must be based on rigorous matheresult of a secondary oxidation of the primary reduc- matical theory before they are of any use as diagnostic tion product by an oxidizing substance (0x2) which is tools. The mathematical methods used in connection not reduced at the cathode under the prevailing Con- with electrochemical reactions which are irreversible in ditions. Polarographers refer to reactions of this class the over-all have become fairly n,ell as "catalytic reactions" and to the currents thereby ~h~ in each case depends upon the rate ,vith produced as "catalytic currents." These polarographic which the participating oxidant is to the electerms are not restricted, howver, to the above case trode and (because the electron-transfer reaction is where Oxz is not reduced elect roc he mi call^ and where usually bidirectional) on the rate with which the rethe first step is fast. ductant is removed from the electrode: it also d e ~ e n d s Class D Ox + ne F? Red (fast) (14~) on the rates of electron transfer (forward and hackRed X (slow) (14b) ward). These rates of supply and removal are in turn dependent on the rates of diffusion and of the This class of reaction is the electrochemical equiva.. "preceeding" or "succeeding" chemical reaction steps. lent of the Conant type of reaction (equation (3)). The latter may be formulated in terms of the cusThe rate-controlling step is a "follow-up reaction" tomary equations expressing the law of mass action which is unimolecular, essentially unidirectional and while the diffusion rates are given by Fick's second homogeneous. The second step may be followed by law. any number of rapid steps without altering the kinetic If it is desired to introduce the electrode potential, relationships involved. the Nernst equation serves if the electron-transfer Diagnostic Criteria. Let us now return to the proh- step is rapid; otherwise, the theory of absolute reaclem of seeking diagnostic criteria for use in differentiat- tion rates is invoked. ing classes of mechanisms. Obviously the class must The equations derived in the manner indicated furbe established before the attempt is made to discover nish valuable diagnostic criteria. Methods have been the more intimate mechanistic details such as the worked out for the calculation of the velocity conchemical formulas for the participating species in stants (kl) for totally irreversible, kinetic, and catalytic each reaction step and the related stoichiometric co- processes. Each class involves a different equation efficients, velocity constants, and free energy changes. and the constants calculated by any given equation There are several methods (not involving the use of must demonstrably be independent of the concentra-

+

-

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JOURNAL OF CHEMICAL EDUCATION

tion if they are to be in accord with the corresponding (56), that the limiting diffusion current in Class D type of mechanism (10). Other diagnostic criteria reactions should not be affected by the L'follow-up" are frequently used. The variation of half-wave po- reaction step but the half-wave potential is thub tential with drop time ( t ) for totally irreversible proces- affected, being smaller than E o and decreasing as the ses must obey the equation: rate constant f o ~the "follow-up reaction" increases (53). These properties of Class D reactiong, although important, do not serve to distinguish them from other mechanistic classes, especially since no where n. is the value for the ratecontrolling step alone method exists, as far as the author is aware, for oh(cf., p. 82 of (10)). The variation of current with the taining reliable E o values for this class of reactions. head of mercury is also frequently used for diagnostic However, Kern (54) also showed that Class D repurposes. I t has been shown for the simplest examples actions have a different dependence on drop time than of classes A, B, and C that when certain parameters "totally irreversible" reactions of Class A, and he used exceed given limiting values in one direction or the this criterion as an argument in favor of classifying other the processes are respectively controlled by dif- the oxidation of ascorbic acid as a Class D reaction fusion or activation and the corresponding type of de- (cf. also (55)). pendence on head of mercury is observed (10). I t has ALTERNATING-CURRENT TECHNIQUES also been shown for class A that a plot of log k, against Introduction. Alternating-current techniques for the E should be linear and that the half-wave potential shifts to more positive values as the temperature in- study of electrode polarization have many attractive features. I n addition to the variables involved in creases. In studying Class C reactions, the so-called catalytic direct-current measurements, frequency, phase angle, current is obtained by subtracting the limiting current, polarization resistance, capacitance, and wave shape observed when Oxg is absent, from the total current become available. With this increased number of observed in the presence of both 0x1 and Ox2 The- variables more aspects of mechanism should become catalytic current is proportional t o the bulk concen- amenable to study. Concentration polarization is much smaller with alternating than with direct curtration of Oxl (16). Class D reactions have not received comparable at- rents and therefore the effects due to slow discharge tention. A few such reactions have been studied are unmasked. Voltammetric curves are not affected polarographically. Smith, Kolthoff, Wawzonek, and by convection because of the short duration of measRuoff (17) studied the oxidation of hydroxychromanes urement. The important coefficient d E / d i is measured and hydroxycoumarans. The primary product of the directly and more accurately as a resistance rather reversible oxidation is transformed irreversibly into a than as the slope of a curve. The bothersome chargquinone which, on reduction, does not regenerate the ing current is involved both in direct-current polaroginitial reactant. Vavrin (18) studied the oxidation of raphy and in alternating-current electrolysis embascorbic acid. He assumed that the decomposition ploying a sinusoidal current but may be eliminated of the oxidized form is a reversible monomolecular re- under appropriate conditions by the use of square action. Oldham (19) studied the reduction of certain waves (40). I t has been the author's experience that an essenmercuro-organic compounds, concluding that the primary reduction product undergoes subsequent reaction tially steady state is achieved very quickly during with hydrogen ions. Kivalo (80) has criticized ad- alternating-current electrolysis. The same conclusion seems to have been reached by Acree, Bennett, Gray, versely the first two of these investigations. A mathematical attack on Class D reactions was and Goldberg (21) in their study of transient phenommade by Kivalo (2O), using the admittedly question- ena in alternating-current electrolytic cells. On the experimental side, alternating-current techable assumption of a steady state. His approach does not yield any knowledge concerning the value of any niques present the added advantage that the cathoderate constant or the concentration of any substance ray oscillograph becomes available for use. Thus a t the electrode surface. It does, however, yield instantaneous values of current and voltage may be equations of the type of equation (9) when suitable measured either as transient or steady state phenomassignments of relative magnitudes of rate constants ena. Polarization Resistance and Capacitance. If an alare made; or it may yield, under certain conditions, equations not meeting the polarographic condition of ternating-current electrolytic cell is used as one leg reversibility, i. e., not satisfying equation (9). The of an impedance bridge, its impedance and the phase chief value of his paper lies in making evident the angle of the impedance may be measured, nothing else. ranges of conditions under which Class D reactions This impedance may be analyzed into its capacitive may be expected to show the polarographic char- and resistive (dissipative) components either in terms acteristics of either reversible or irreversible reactions. of a series or parallel combination of a resistor and Kern (53, 54) has shown mathematically, in com- capacitor. Thus if one speaks of the equivalent series plete agreement with the earlier work of Koutecky resistance and capacitance of an electrolytic cell, he

VOLUME 33, NO. 11, NOVEMBER, 1956

means that a resistor and capacitor having respectively these values and placed in series in the balancing arm of the bridge would have the same impedance and cause the same phase shift as the cell does at the given frequency (11). For some purposes it is convenient to use alternating current alone; for other purposes it is better t o use a very small alternating current superimposed on a relatively large direct current. I n this case the direct component sets the voltage a t which it is desired to work and the alternating component makes possible measurements of the impedance. It is desirable to use a very small alternating current so that the wave shape is not distorted and mathematical equations developed on the basis of a specified wave shape may be used. If alternating current alone is used, the attainment of the desired voltage usually results in a high current and wave distortion. An exception is found in the use of poised solutions of reversible redox systems (22,2!3). It is customary to use one electrode of very large surface area so that its impedance is negligible. The cell impedance may then be taken as the impedance of the other electrode alone. Consideration may next be given to the origin of cell capacitance. If the cell electrolyte contains ions which are not readily discharged and a sufficientlylow potential is applied across the electrodes, no electrontransfer reaction occurs and the electrodes are said to be "ideal polarized electrodes." Under these conditions, ions are adsorbed on the electrode's surface. There are thus produced two layers of electrical charges (one on the electrode, the other on the adsorbed ions), the so-called electrical double layer which functions as a capacitor when a potential is applied across it. The current which is used in charging the double layer is called the "charging current" and the associated capacitance is known as the double layer capacitance, Cd, which can be measured with an impedance bridge (24, 25). Unlike an ordinary physical capacitance, Cd is a function of the voltage. Its magnitude usually lies between 5 and 90 pFcm.? (26). If conditions are such that chemical transformation is produced by the applied voltage, the current used chemically is called the "faradaic current." Under these conditions there is a partition of the total current between the faradaic and charging processes; the double layer functions as a very leaky dielectric and the capacitance increases enormously, sometimes reaching values in excess of 1000 pF/cm.% Since the double layer is not discharged, Cb must still he considered as a component of the capacitance. The other component has been named the "pseudocapacity" by Grahame (25). He demonstrated its chemical nature by showing that the half-wave potential for cadmium chloride coincides with the maximum of the capacity-voltage curve. The pseudocapacity arises from the concentration polarization which in turn is conditioned by diffusion. During one half-cycle electrochemical trans-

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formation a t the electrode results in the chemical storage of energy and the accompanying creation of concentration-overvoltage. During the succeeding half-cycle the transformation is reversed and the energy is returned to the external circuit. Thus the electrolytic cell simulates a physical condenser which stores energy on one half-cycle during a voltage rise and restores it to the external circuit during the next half-cycle. I n both cases the voltage lags behind the current by 90' and we say that a polarization capacitance is involved. A faradaic process which is accompanied by a negligible reverse current will, of course, give rise to no pseudocapacity (27). To the extent that diffusion of reaction products away from the electrode prevents reversal of the electrochemical reaction with current reversal, the polarization capacitance will be decreased (reactance and impedance increased). The same thing would result from any lag in the diffusion of the reactants toward the electrode. Thus polarization capacitance becomes a function of diffusion rates or, more accurately, of the relation between diffusion rates and the period of the alternating current, I t thus becomes a function of the frequency of the alternating current, i. e., it shows dispersion. Polarization resistance was discovered in the year 1896 by Wien (28). He observed that alternatingcurrent electrolysis produced a resistance increase (AR) over the "true" resistance (R,) of the cell, i. e., over the resistance calculated from cell dimensions and the conductance of the electrolyte. Thus one may write: R = RT AR (17)

+

He suggested that this polarization resistance, AR, might be caused in part by the formation of poorly conducting surface layers, in which case the accompanying electrical energy loss (characteristic of electrical resistance) appears as heat; or the electrical energy could be irreversibly stored as chemical energy and hence, being lost to the electric circuit, the "loss" would he manifested as a resistance. He envisioned the cause of this second kind of polarization resistance as a "spontaneous depolarization." As examples of "spontaneous depolarization" he mentioned unidirectional secondary reaction steps and diffusion of primary reaction products away from the electrode with the result that the electrode reaction could not undergo reversal and restore energy to the external circuit on current reversal. Modern research has shown Wien's surmises t o he correct. An electrode layer resistance can be produced by electrolysis in certain cases and diffusion waves definitely do give rise to a component of the resistance which is called "diffusion resistance" by the Germans. However, as far as the author is aware, nobody has yet definitely established that unidirectional secondary reaction steps contribute to the polarization resistance although surely such must be the case. Modern research has, moreover, definitely

JOURNAL OF CHEMICAL EDUCATION

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established that slow electron transfer at an electrode gives rise t o a component of the polarization resistance. This component is called "transition resistance" by the Germans (e. g., 29) and is symbolized as 8 by Grahame (11). The modern point of view with regard to the relationships existing between the various components of the resistance and capacitance can be conveniently described by the equivalent circuit for a half-cell shown in Figure 1. If R, is removed from Figure 1, the resulting equivalent circuit will be found to be in complete agreement with those suggested by Randles (SO), Rozental and Ershler (31), Gerischer (Sg), and Grahame (11). Grahame's symbols are used in the figure. W stands for the "Warburg impedance," which might be called the diffusion impedance. Al-

DOUBLE LAYER BRANCH

-"\-+.MR,

FARADAIC B R A N C H F

i R,

taih of the theory (23, 29, 31, 37, 38). Unless othe1wise indicated, Grahame's treatment will be followed. The transfer resistance, 0, is a function of the electrontransfer reaction whereas the capacitive and resistive components of the Warburg impedance are expressed, respectively, as the sums of a number of capacitances and resistances, each of which is a function of one substance; i, e., one resistance and one capacitance may be written for each substance which is a component of the electrolyte (in the sense of the phase rule). Grahame groups substances in classes. It will be convenient to discuvs first systems containing only substances of Class I (supporting electrolyte) and Class I1 (substances which undergo periodic changes of concentration a t the electrode as the result of electrochemical reaction, but which are not involved in ( I preceeding" or "succeeding" reactions of any kind). The following equations are of the most interest. They were derived from fundamental principles and apply equally well when alternating current of frequency w (radians sec. -I) is used alone or when it is superimposed on a direct current. The series equivalent capacitance of the faradaic branch is defined as:

" 1. Equivalent C i x v i t

though it may be expressed as a series combination of where : a diffusion resistance and a diffusion capacitance, Grahame prefers not to use these conventional circuit elements in the picture because the properties of the and : Warburg impedance are not those of any conceivable network built up of physical capacitors and resistors. I t is a new type of circuit element and deserves a new symbol. v t is the number of equivalents of an ion of the ith kind The electrode layer resistance, R,, was added t o produced by chemical action when one faraday of elecGrahame's picture by Remick and McCormick (23) tricity crosses the phase interface a t the electrode. because it was demanded by their experimental results using platinum electrodes in a poised ferrocya- The series equivalent resistance is defined as: nide-ferricyanide solution. I t apparently is equal to zero in many other cases. Vetter (38) also saw fit to include as a possibility a "protective-layer resistance" Since u involves a summation, it is obvious that in his equivalent circuit. C,* and R,* represent total contributions from all Mathematical Theory of the Faradaic Admittance. substances involved. From the standpoint of reaction mechanism studies, From equation (21) it is clear that if R,* is plotted it is only the faradaic branch of Figure 1 in which we against l/w"', a straight line results whose intercept are interested. In order to calculate the faradaic im- on the R,* axis is 8. Thus a simple means of depedance (or admittance, which is the reciprocal of the termining 0 is available. impedance) it is necessary t o make vectorial subtracIt follows simply from equations (18) and (21) and tions of the double layer capacity Cd, the electrolytic from the definition of reactance, X, that the resistive resistance of the cell R,, and the electrode layer re- and reactive components of the Warbnrg impedance sistance R,. Of these, Cd and RT can be determined are equal, i. e.: by well-known methods (25, 55, 59). The same investigators mentioned above in relation to the equivalent circuit (11, 30, 31, 32) have also succeeded in developing from basic principles a matheThe exchange current (io) may readily be calculated matical theory which extends the earlier theory de- from 8 by the following equation (SO, 42) : veloped by Warburg (54), Kruger (35), and Rosebrugh and Lash Miller (56). Only a few of their most significant conclusions can be mentioned here. A variety of experimental tests have substantiated the main deThe exchange current may further be related to

VOLUME 33, NO. 11, NOVEMBER, 1956

the rate constant of the electron transfer step: The rate constant, A is the electrode area in k,, is related to those previously used in equations (4) and (5) by the defining equation:

where E,O may usually he taken to be the standard Nernst potential although strictly speaking it is not identical with the Nernst constant (cf., (lo), p. 35). It should be clear from equations (6a), (6b), and (25) that k , is the value of either k , or ka when E has the particular value ECo, at which value equilibrium is achieved. The phase angle +* is the angle by which the current through the faradaic branch leads the corresponding voltage. By definition, tan +* = X,*/R,*, or: reaction falls in any given mechanistic class, the mathematical requirements of that mechanism must first be Since X,* = R,* when 9 = 0, i t is clear that the phase deduced from basic principles (often an imposing angle approaches 45" as the rate of electron transfer task!) and then the reaction must he shown experiapproaches infinity. mentally to be in accord with these requirements. Diagnostic Cm'teria. Owing to the relatively small I n working toward this goal, Grahame (11) has conconcentration polarization obtaining when alternating sidered a number of different classes of substances and currents are used, 0 and hence k , (cf., equations (23) and has shown that Figure 5 conveniently represents the (24)) may be determined for electrode processes which faradaic branch of the equivalent circuit for reactions are too rapid to be studied by other methods. Thus involving substances other than Class I or I1 (or in Randles (22) measured the values of k, for a number of addition to them). I t will be noticed that an addi"reversible" redox systems. He found, for example, tional impedance, X, has been added. This is not a that the rate constant for the transformation of ferricy- conventional circuit element and rill have different anide to ferrocyanide is about 20 times as fast as the mathematical properties for each different class of transformation of ferric to ferrous ion although both substances. systems behave reversibly as judged by the potentioAlthough emphasis has been given to the problem metric criterion. When 0 is very small, the reactiou is of determining in what mechanistic class a given rediffusion-controlled and the phase angle is 45'. Under action falls, it must not be forgotten that this is only these circumstances, equation (26) shows that the one step toward the goal. Another vitally important quantity wC,*R,* should equal unity at every frequency. problem is the determination of the chemical formulas This is a convenient test. If 0 is appreciably large, it is still true that the resistive and capacitive components of the Warburg impedance are equal. Consequently the diagnostic equation becomes:

,c,*(n,* - a ) = 1

(27)

Equation (27) is diagnostic for Class A reactions. Furthermore, the phase angle is less than 45'. Another diagnostic test for Class A reactions is furnished by equations (18) and (21) which give rise to the linear relationships pictured in Figure 2. Gerischer (29) has shown that the dispersion curves of Figure 2 are replaced by those of Figure 3 if there is a homogeneous "preceeding" step (Class B) and by Figure 4 if the "preceeding" step is heterogeneous. The differences in Figures 2 , 3, and 4 constitute powerful diagnostic criteria. As far as the author is aware, no alternating-current studies have yet been made on reactions of Classes C and D. In general it can be said that in order t o demonstrate that the mechanism of an electrochemical

Fiwr.3. Di.*-ion

Curves for C1.a.

B ReactLon.

(H.mog.n.ou.)

572

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The values of the differential coefficients on the lefthand sides of equations (29-32) can be obtained from the slopes of the linear curves obtained when In io is plotted against ro (the equilibrium potential for the gross process), A, or B. The term blnB/bro, or its reciprocal, can also be determined directly from experiment and, in the presence of a sufficient excess of supporting electrolyte, blnfx/blnCx may be taken as zero. If, then, 9 is determined by the extrapolation method already described and the exchange current then calculated by equation (23), n being known, 0 can be obtained from either equation (29) or (31). Finally, substitution of this value of a into either equation (30) or (32) makes possible the evaluation of the stoichiometric coefficient., v., of the electrootransfer step (not necessarily the slowest step). FurF i w m 4. Dispa-ion Curves for ClB Reactions Wstarowneous) thermore, since v and n are now known, the chemical formula and charge type of the complex ion of the participating ionic and molecular species in each [MX,1"-")- , which is the participating species in the step and their stoichiometric coefficients. Notable electron-transfer step, may be determined. efforts in this direction have been made by Vetter Gerischer's experiments indicated that the value Gerischer (45), and Parsons (46). Only of a is apt not to be very close to (58, 4.9, Thus in the case Gerischer's approach will be reviewed here. of zinc ion complexes in oxalate solution, he found that Gerischer developed mathematical equations for the a = 0.75 0.05. It should be emphasized, incidischarge of metallic ions from solutions in which they dentally, that this method is the most accurate one exist as complexes. The "preceeding" step and the yet devised for the measurement of a and that Gerisdischarge step may be written as follows: cher's values of 0 agree well with those of Randles and [MX.](a-n) - e [ M X . ] ( Y - ~-) + (s - v)X(2&) Somerton (22). Another point of incidental interest [MX,](p-=)- + ne M + vX(28b) in relation to Gerischer's work with various zinc ion complexes is that in no case did the dispersion curves show the presence of slow "preceeding" reactions, i. e., curves like those in Figure 3 were not obtained. He concluded that the "preceeding" reactions involved must be very fast. The value of a serves as a diagnostic criterion in the Fi8. Faradaic Branch for R.astion. Involving sense that values less than unity can only be accounted 1 and I1 SubstanOther Than G~ahama'.Cl-n for in terms of a slow discharge step (27). (OF i n Addition to Them) Cathode-ray Patterns. One of the most enticing potentialities of alternating-current techniques is the Owing to the difficulty of determining the concentra- study of instantaneous variations of the current and tion of any one ionic species, he used the quantities of electrolytic cells as shown on the screen of a A and B which represent respectively the total bulk voltage cathode-ray oscillograph. The cathode-ray patterns concentrations of all ions containing the metal ions, (oscillograms) may picture the relation between M+", or the complexing anions, X-. He then de- voltage and current, or the time variation of either veloped the mathematics on the assumptions that voltage or current may be shown. Either "steady B>A and that a supporting electrolyte was used. state" conditions or transients may be studied, the His most pertinent equations are the following in latter being employed in discharge experiments. which the subscript, "Var.. ." signifies "change of Oscillographic polarography utilizes the oscillograph . ..alone with all other concentrations constant." to produce polarograms. The introduction of harThe concentration of the metal could be variable if, for monics into alternating current waves (i. e., wave example, the electrode were an amalgam. distortion) can be studied oscillographically. Apparently Reichenstein (47) was the first t o employ an oscillograph in studying the polarization of electrodes. His apparatus was crude and therefore his conclusions are questionable (48, 49). A particularly valuable use of the oscillograph is in the study of electrode-layer resistance. When external sources of e.m.f. are removed, that portion of the electrode polarization due to an I R drop through the electrode layer drops immediately to zero leaving

a),

*

=

VOLUME 33, NO. 11, NOVEMBER, 1956

the remaining polarization to decay slowly. This phenomenon is clearly evident on the oscillogram. This technique has been reviewed and furthered by Ferguson (50). Silverman and Remick (L.9) ~. , made oscilloeranhic studies for the purpose of exploring the potentialities of this technique as a tool for reaction mechanism studies. Their approach was largely qualitative but perhaps several of their conclusions are worth mentioning. Their oscillograms were voltage-current characteristics in which the voltage was the potential difference between one of the working electrodes and a probing electrode while the current was that which passed between the two working electrodes. As a guide for the qualitative interpretation of the oscillograms, they showed that the convex portions (the point of observation being considered to be outside of the oscillographic figure) of the oscillograms may occur when d q / d E is decreasing, is constant, or is increasing dightly whereas concave portions may occur only when d q / d E is increasing rapidly. Remembering that d q / d E is the differential capacitance and that the double u

.

redox system, it is apparent that during those portions of a cycle in which most of the electricity is being used to charge the double layer, the oscillogram will be convex and the voltage rise great. On the other hand, when the current is being diverted from charging the double layer to causing chemical reaction, the oscillogram will become concave and the voltage rise will be comparatively small. These features are easily recognizable in Figure 6, which was constructed from their data. During a given infinitesimal period of time there should be a partition of the electricity between these two processes (i. e . , charging the double layer and causing chemical reaction) in proportion to the values of the corresponding capacities Cd and C,*. If one considers a typical Nernst S-curve and disregards diffusion, it is apparent that at the two ends of the curve d E / d q is terrifically large and therefore d q / d E terrifically small and most of the current, will be used in charging the double layer. The reverse situation obtains when the redox ratio at the electrode surface corresponds t o the middle, poised region of the Nernst curve. Indeed, here one might expect that the voltage (corrected for the IR, drop through the solution) would remain almost constant if the current density is not too high. In Figure 6 such a region of constant voltage is observed from E to A to C. The poised region is entered a t E and oxidation continues, a t a decreasing rate, from A to B. At B the current reverses and the.ferricyanide produced by oxidation and not yet lost by diffusion is reduced again, the end of the poised region being traversed again a t C. (The fact that EAB > BC is presumably the result of diffusion loss.) Beyond C most of the electricity presumably goes into charging the double layer although some of it probably is used in discharging hydrogen ions, a process for which

Figule 6.

Currant-Voltage Chaxacteristics Corlected for IRT

Electrolyte: 0.5 M aqueoua K.Fe(CN)a; current density: cm.-*; frequency: 100 c.p...

88.8 r.m.s.

ma.

the pseudocapacity is small because of the slow discharge rate. From D to E the double layer discharge is presumably accompanied by the oxidation of hydrogen and at E the potential has become sufficiently anodic so that the poised region is entered again. They found, as expected, that when the solution initially contained only ferrocyanide, the linear portion of the figure occurred a t the top of the diagram. When the initial solution consisted of an equimolar mixture of ferrocyanide and ferricyanide, the oscillogram was a straight line (or possibly an exceedingly thin ellipse) showing that the system never left the poised region during the course of the electrolysis. Silverman and Remick were somewhat mystified when an equimolar ferrous-ferric solution yielded a badly distorted ellipse with both anodic and cathodic tails instead of yielding a straight line. However, in view of the subsequent demonstration by Randles and Somerton (2.2) that the ferrous-ferric system is much more sluggish in its redox behavior than is the ferrocyanide-ferricyanide system, the appearance of the ferrous-ferric oscillogram becomes understandable. It is apparent from the foregoing discussion that it is possible to learn much qualitatively from cathode-ray patterns. There are also unexplored possibilities for gaining quantitative information of value. The qnalitative information thus quickly obtainable can he of very great practical advantage to an investigator attempting to pick out a suitable system for quantitative study. Van Cakenberghe (51) has devised a clever oscillographic technique by means of which the transfer

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coefficient may be calculated. He showed mathematically that a t the half-wave potential of a reversible reaction, with ol = 0.5, the second harmonic of the distorted wave should disappear. If the discharge step is moderately slow, with ol different from 0.5, the potential at which the second harmonic disappears is not the half-wave potential. However, at this potential the transfer coefficient bears a simple relation t o concentrations and can be calculated. His experimental method involves the polarization of the electrode by an adjustable constant voltage and the superimposition of a small alternating voltage. The cell current is led through a resistor across whose terminals a voltage is developed. The alternating component of this voltage is amplified and then passes through a filter, tuned t o the second harmonic, to the oscillograph. The polarizing voltage is then adjusted until the oscillograph indicates that the harmonic has disappeared. OTHER METHODS

It is doubtful if the avowed purpose of this review would be well served if it were extended to the formidable length required to include all phases of electrochemistry pertinent to electrode mechanisms even though the discussion were continued on the chosen vlane of suverficialitv. Admittedlv the effects of adSorption shbuld be biscussed b u t U a reasonably adequate discussion of this subject would add many pages. Among the methods omitted from discussion should be mentioned voltammetry at constant or controlled current or with continuously changing potential and methods involving rapid rotation of electrodes or stirring of solutions (agitation is advantageous because a steady state can then be achieved). These methods are reviewed critically and in great detail in Delahay's book (10). Finally it should be mentioned that measurements of current efficiency, although less glamorous than polarographic and impedance measurements, are capable of yielding information of genuine value for reaction mechanism studies. This is particularly true when several reaction products are formed as in the Kolbe electrochemical synthesis (for example (5%)). The use of alternating currents in this field of research has not been sufficientlyexploited. LITERATURE CITED (1) PETERS,R., Zeit. phys. Chem., 26, 193 (1898). (2) CLARK, W. M., Studie~on Oxidation-Reduction, I-X, U. S. Public Health Service, Washington, D. C., 1928. (3) MICHAELIS, L., Chem. Reus., 16, 243 11935). (4) ELEMA, B., Rer. trav. chim., 50, 907 (1931). (5) REMICK, A. E., J. Am. Chem. Soc., 58, 733 (1936). (6) IZORTUM, G., AND J. O'M. BOCKRIS, ?Textbook of Electrochemistry," Elsevier Publishing Co., 1951, p. 281. (7) CONANT, J. B., Chem. Revs., 3, 1 (1926). A,, ET AL., Ber., 62B, 1942 (1929); Ann., (8) WEISSBERGER, 481, 68 (1930); Be?., 64B, 1200 (1931); 65B, 1815 (1932); J. Chem. Soc., 1935, 223, 226. (9) GLASSTONE, S., K. J . LNDLER,A N D H. EYRING, "The Theory

of Rate Processes," McGraw-Hill Book Co., Inc., New York, 1941.

(10) DELIHAY,P., "New Instrumental Methods in Eleetro(11) (12) (13) (14) (15) (16)

chemistry," Interscience Publishers, Inc., New York, 1054 D. C., J. E e c t r o h . Soc., 99, 370C (1952). BUTLER,J. A. V., "Electrical Phenomena at Interfaces," The MrtomiIlan Co., New York, 1951, p. 157. KOLTHOFF, I. M., AND J. J. LINGANE,"Polarogmphy," Interscience Publishers, Inc., New York, 1952, p. 234. YEAGER, E., T. S. OEY,and F. HOVORKA, J. Phys. Chem., 57,268 (1953). R. PENN,E. YEIGER, AND F.HOVORKA, Tech. Rept. No. 16 to O.N.R. (1955). KIVALO,P., K. B. OLDHAM, AND H. A. LAITINEN, J. Am. Chem. Soe., 75, 4148 (1953). DELAHAY, P., AND G. L. STIEHL,J. Am. Chen~.Soe., 74,3500

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(1952). (17) SMITE, L. I., I. M., KOLTHOFF, S. WAWZONEK, AND P. M. RUOFF,J . Am. Chem. Soe., 63, 1018 (1941). Z., Collection Czechoslov. Chem. Communs., 14, 367 (18) VAVRIN, (1949). (19) OLDHAM, K. B., Thesis, University of Manchester, England, 1952. (20) KIVAM,P., Aeta Chem. Sand., 9, 221 (1955). (21) ACREE,S. F., E. BENNETT,G. H. GRAY,AND H. GOLDBERG, J . Pnys. Chem., 42, 871 (1938). (22) RANDLES, J. E. B., AND X. W. SOMERTON, Trans. Faraday Soc., 48, 937 (1952). (23) REMIOK, A. E., AND H. W. MCCORMICK, J Electrochem. Soe., 102, 534 (1955). (24) PROSKURNIN, M., A N D A. N. FRUMKIN,Trans. Foraday Sac., 31, 110 (1935). J. Am. Chem. Soc., 63, 1207 (1941). (25) GRAHAME, (26) GRAHAME, Chem. Revs., 41, 441 (1947). (27) GRAHAME, Ann. Rev. Phys. Chem., 6, 337 (1955). (28) WIEN,M., Ann. Phyaik. u. Chem., 58, 37 (1896). (29) GERISCHER, H., Z. Elektmchem., 55, 98 (1951). (30) RANDLES, J. E. B., Dism~8imsFamday Soe., 1, 11 (1947). (31) ROZENTAI,, K., A N D ERSHLER, B., Zhw. Fiz. Khim., 22,1344 (1948). (32) GERISCHER, 2.Physik. Chem., 198, 286 (1951). (33) FRUMKIN, A,, AND M. PROBKURNIN, Trans. Paraday Soc., 31, 110 (1935). (34) WARBURG, E., Ann. Physik. U. Chem., 67, 493 (1890). F., Z. physik. Chem., 45, 1 (1903). (35) KRUGER, (36) Rossenuce, T. R., AND W. LASHMILLER,J . Phy8. Chem., 14, 816 (1910). (37) RLNDLES, Trans. Faraday Soe., 48, 828, 937 (1952). (38) VETTER,K. J., Z. phys. Chem., 199, 285 (1952). 139) JONES. G.. A N D S. M. CHRISTIAN, J. Am. Chem. Soc., 57,272 (1935). (40) BARKER, G. C., AND I. L.JENKINS, Analyst, 77, 685 (1952). (41) Bacmls, J. O'M., "Modern Aspects of Electrochemistry," Academic Press, Inc., New York, 1954, Chap. 4. (42) ERSCHLER, B., Di8cu~sion8Faraday Soe., 1, 269 (1947); Zhur. Fia. Khim., 22, 683 (1948). (43) VETTER, Z. Elektrochem., 55, 121 (1951). 1441 K. J.. A N D G. MANECKE. . 195, 270 . Z. .P ~. V S Chem.. ~, VETTER. (195oj. (45) GERISCHER, Z. physik. Chem., 202, 292, 302 (1953). (46) PARSONS, R., Tmns. Paraday Soe., 47, 1332 (1951). (47) REICAENSTEIN, D., Z . Elektroehem., 15, 913 (1909). (48) FURTH,A,, Z . Elektrochem., 32, 467 (1926). (49) SILVERMAN, J., AND A. E. REMICK, J. Elect~ochem.Sac., 97, 335 (1950). (50) FERGUPON, A. L., Trans. Eleet~.ochem.Soe., 76, 113 (1939). (51) VANCAKEENBRGHE, J., Bull. me. h i m . Belges, 60, 3 (3951). (52) THIESSEN, G. W., Ill. A?d. Sci. Trans., 43, 77 (1950). (53) KERN,M. H., J. Am. Chem. Soc., 75, 2473 (1953). (54) KERN,ibid., 76, 1011 (1954). (55) KERN,ibid., 76, 4234 (1954). (56) KOUTECKY, J., Colleelion Czech. Chem. Communs., 18, 183 (1953).