B. E. Conway and Mark Salomonl University of Ottawa ottowo 2, Canada
Electrochemistry:
In Teaching Physical Chemistry
A t certain periods in the development of a subject i t is sometimes desirable to examine, on the one hand, the historical progress of that subject, and on the other, its orientation with regard to kindred fields of research. This is particularly true of the subject of electrochemistry, which in the last one to two decades has seen relatively rapid advances both in the quantitative theoretical formulation of a number of fields within its discipline and in the striking progress made in its applications. Recognition of these developments and changes of orientation of the subject, in relation to the historical background of the field, are essential in the teaching and presentation of electrochemistry from a modern and integrated point of view. These advances, however, appear, as yet, to be not a t all widely known outside what is commonly (hut incorrectly-see below) regarded as the "specialized field" of electrochemistry: in fact, in the teaching of undergraduate physical chemistry usually rather little is taught (if a t all) outside the classical areas of electrochemistry associated with reversible cells and the thermodynamic information that can be derived from their study, together with wellknown problems of ionic solutions such as the interpretation of activity coefficients and conductance behavior through the theories of Dehye and Hiickel (1) and Onsager (g). To an appreciable extent, this orientation in the pedagogy of the subject is due to the inadequate and anachronistic presentation of electrochemistry in many general physical chemistry textbooks, both old and modern, which are currently recommended to the undergraduate student. In the present article, we shall examine to what extent the subject has broader ramifications and newer aspects, and how these may be related to other more commonly developed fields of physical chemistry, and at the same time how modern developments in the field may be presented in relation to historically important and interesting landmarks of the subject. The way in which the remarks to he presented below will pertain to "chemistry curricula" will very much depend on the particular university or college concerned. At the moment, in the U.S., the number of semesters devoted to physical chemistry courses varies a great deal, and only a relatively small number of universities give a course specifically in electrochemistry a t the undergraduate level. There are also differences of a more general kind between curricula in the U.S. and Canada, where, in a number of universities, an honors degree program is offered, in the last year of which specialized courses in such topics as advanced theory of kinetics, electrochemistry, quantum chemistry, stereo-
'
Sprague Research Fellow at the University of Ottawa, 196264. Present Address: Department of Chemistry, Rutgers University, New Brunswick, N.J.
554
/
Journal o f Chemical Educofion
Its Role
chemistry, and statistical thermodynamics are given. However, in those cases where a more limited undergraduate program is involved, most of the remarks made below can be regarded as applying just as well to the initial stages of subsequent graduate instruction in physical chemistry and electrochemistry. I t is felt by the writers that much is to he gained by the presentation of electrochemistry as one whole subject, i.e., one should aim a t inclusion in one course of both the solution (ionic interaction, solvation, dissociation) and the interfacial (electrode processes, doublelayer theory) aspects of the subject. While the interfacial aspects of the subject are the ones in which the newest and the most exciting ground is being broken (see below), it is of interest to note that in 1965, for example, over 250 new papers on aspects of solutim electrochemistry were published. I n a number of institutions and in many textbooks, aspects of the behavior of ionic solutions are included in the "physical chemistry of liquids and solutions," and the significanceof reversible electromotive forces are treated separately under "thermodynamics." The important and close relations between these aspects of the subject and those concerned with the doublolayer and electrode processes (which often are less fully treated) thus tend to be obscured or are not sufficiently apparent to the average student. For example, the fundamental factors determining the emf of a reversible half-cell element are not usually presented, and such matters as the role of electronic work function, contact p.d. in the external circuit, solvation free energy of the reactant ions, etc., and the kinetic treatment of electrochemical equilibrium tend to he neglected with a resultant lack of understanding by the student of the origin of emf. The Structure of Electrochemistry and its Historical Development
I n the early development of quantitative physical chemistry, and indeed in regard to the question of chemical constitutiou itself, electrochemistry played a rather central role; with regard to electrical aspects of the constitution of matter, studies of electrolysis of solutions by Nicholson (5), Faraday (4), Berzelius ( 5 ) , and Daniel (6) provided information complementary to that later obtained by Thomson and by Crookes on the nature and products of electrical discharge in gases. These discoveries led ultimately to an understanding of the nature of chemical combination, and, through the appr* ciation of the nature of metals by Riecke, Drude, and Sommerfeld, to a molecular understanding of the mechanism of electrolysis. This early work together with that of Ostwald (7), Arrhenius (8),and Kohlrausch (9) established the nature of ions and the special properties of electrolytic solutions.
I t is of interest to recall how early it was that electrical interactions were recognized as a basic factor in chemistry. Explicit ideas on this subject had been formulated by Priestley in 1767, and later by Berzelius; and in 1809, Jane Marcet in her two volumes quaintly titled "Conversations in Chemistry," was able to say of Davy's ideas: . . . . he supposes that there are two k i d s of electricity, with one or other of which all bodies are united. These we distinguish by the names of positive and negative electricity: those bodies are disposed to combine which possess opposite electricities, as they are brought together by the attraction which these electricities have for each other.
Continuing she wrote, most significantly: But whether the hypothesis he altogether founded on truth or not, it is impossible to question the great influence of electricity in chemieal combinations.
It must be recalled, however, that the idea of free ions existing independently of the passage of current or of the applied voltage was not at that time developed, and the significance of the energy of hydration of ions in relation to the problem of the degree of dissociation of salts in solution was not well understood in Faraday's time nor indeed until after the work of Arrhenius on the conductance behavior of electrolytes where the "intrinsic" salts, now recognized as "strong electrolytes" in aqueous medium, were still regarded as incompletely dissociated to an appreciable extent. This early work led, however, through Bjerrum's ideas (lo), to the distinction of "nonideality" arising from incomplete dissociation from that which can be regarded (more properly) as arising from long range ion-ion, short range ion-solvent, and excluded volume effects. In fact, the first quantitative theory of non-ideality of electrolyte solutions was that derived by Milner (11) (for the freezing-point depression for 1:1 salts) in 1912-13 and in a more elegant and mathematically less complex form by Debye and Hiickel (1) in 1923for the electrochemical behavior of ions in solution, and subsequently extended by various authors. It is not often noted, however, that the type of mathe matical treatment developed in the DebyeHiickel theory (use of an electrostatic energy function in a Boltzmann distribution, linearized to a first approximation, in conjection with Poisson's equation) for a three-dimensional spherical distribution was already formulated in 1913 by Chapman (12) for the electrical doublelayer a t chargedinterfaces, discussedfirst by Gouy (lS),where an analogous one-dimensional ion distribution arises. In parallel with these early developments of the theory of electrolyte solutions and the experimental measurements of osmotic pressures, conductance behavior, and activity coefficients (1, 2, 7, 14), was the empirical formulation of the first basis for the study of electrode processes (apart from the quantitative laws of Faraday) by Tafel in 1905 (15); although a surprising variety of both inorganic and organic electrochemical syntheses had been performed by the end of the 19th century, the mechanisms of most of these have only recently become understood, and the theoretical basis of the kinetics of electrode processes had yet to be formulated in a fundamental and basic manner by Gurney (16), Baars (17), Erdey-Gruz and Volmer (la), Frumkin (19), and Butler (20) in the early 1930's. The essential kinetic formulation of the forward and backward steps in electrode process equilibria was, however, presented
first by Butler (21) in 1923 but not quite in the finally accepted general form given by Erdey-Gruz and Volmer (18) for a single ion discharge step. In considering the structure (22) of the subject of electrochemistry, the root of the subject must be associated with the electrochemical recognition of ions and their relation to the particulate electrical nature of the constitution of matter first appreciated by von Helmholtz, Johnstone-Stoney, and Thomson. From this starting point, it is useful first to distinguish the fields of study of ionic solutions and ionic, fused salt melts which constitute the subject of "ionics," as it may be called (23). From the study of the electrical nature of matter stems the field of "dielectrics," a subject more properly in the domain of general physical chemistry and which is directly related to the question of the energetics of solvation of ions through polarization of the solvent dielectric. In the field of ionics, it is useful to recognize two broad subdivisions: one concerned with the thermodynamics of salts in solution or in the fused state and the behavior of reversible cells with regard to the electrochemical potentials associated with potential-determining ionic reactions; and the other concerned with dynamic properties of ions in solution, viz., conductance, viscosity, transference numbers, and diiusion. It is useful to stress that the dynamic properties at finite eoncentrations of salt are determined in part by the same kind of interaction effects which determine the activity of an electrolyte and related derived properties such as the relative standard partial molal entropy and heat content of the electrolyte. Correspondingly, at infinite dilution, the dynamic properties are determined only by the properties of single ions and their interaction with the solvent. In addition to what we have termed the dynamic properties of ions, a large body of modem electrochemistry is concerned with the kinetics of electrochemical reactions in which ions are adsorbed and discharged or produced by electrolysis, and here individual ionic reactions can, of course, by studied a t appropriate anodes and cathodes. This field of electrochemistry forms the second, and perhaps contemporarily the most important, main division of the subject, and may conveniently be called 'Lelectrodics"as suggested elsewhere (?AS). The main sub-divisions of the subject may therefore be written as: Electrochemistry Ionics (The study of ions in [Dieleotrics] solution and in fused salts and their interactions among themselves and with solvents. The role of ionic processes in biological systems.)
Eleotrodics (Study of kinetics of discharge and ionization reactiom; eleetrocatalysis at interfaces and electrochemical adsorption.)
and the relationships between the various areas of study in these sub-divisions may be shown as the "Structure of Electrochemistry" in the more detailed scheme given in the table. Ionic Solutions
The students' introduction to the physical chemistry of solutions often begins by reference to ionic solutions, e.g., with regard to osmotic behavior and solvent vapor Volume 44, Number 10, October 1967
/
555
Structure of Electrochemistry (from ref. (221, by permission of the publishers, Ronald Press Company).
pressure lowering. This, however, is probably not the best way of discussing elementary aspects of ideal or non-ideal solutions, since the non-ideality of electrolyte solutions arises in a rather complicated way from a mixture of ion-solvent effects, ion-ion effects, and from incomplete dissociation effects (e.g., in historically considered cases and with certain weaker electrolytes, organic salts, and acids, and with 2:2 and2:3 or 3:2inorganic salts). Great emphasis is usually placed on the 556
/
Journal of Chemical Education
Debye-Hiickel theory of electrolytes but the historical and physical significance of quasi-lattice theories (cf. more modern lattice theories of simple binary solutions and polymer solutions) such as that of Ghosh (27)) leading to a cube-root law for conductivity is rarely mentioned. This is undoubtedly because the Debye-Hiickel theory in its "limiting law" form, and the corresponding theory for equivalent conductance, can provide a firm anchor for the undergraduate student (and perhaps the
profcsror) in his jourt.ey through the sue(:w4ot1o i g(md, \,ad, or indifferent approxirnntit~l~s r o \\.hie11heis eqxsed in vmious t h w ~ w i r ~deri\.,~tims il i l l nltviicul ehcmi~trv. " However, such an outlook may be deceptive, since in the derivation of the limiting law, the Boltzmann exponential distribution is really replaced, as the familiar "Debye-Huckel approximation," by a linear distribution expression, for example A
becomes where ni,+is the number of ions per unit volume in an element of solution, II., the potential, and no,the average bulk concentration of ions. The simplification in eqn. (2), which leads to a convenient solution for fi from Poisson's equation, is thus purchased by the introduction of some physical inconsistency and a degree of physical unreality in the distribution equation (28, 29). Thus a lack of physical reality of the Debye-Huckel theory for anything but the most dilute solutions (< ca. 0.001 M ) has been illustrated by Frank and Thompson (28) by calculation of the number of ions in various spherical shell elements around the "central reference ion" treated in the DebyeHuckel theory. This calculation involved estimation of what fraction of the total effect of the ionic atmosphere on the central ion can be associated with what numbers of ions in these shell e l e ments of successively greater size (say up to a radius of 1 / ~the , effectivelimit of the ionic atmosphere a t a concentration determined by K % = (4ne2/ckT)2n&). Except for very dilute solutions, these calculations lead to far too "discrete" an ionic atmosphere to be consistent with the continuous space charge distribution of density p , involved implicitly in the use of the Poisson equation for obtaining the potential J.. For strong solutions, (>0.2 M), a quasi-lattice distribution (27) of ions with nearest-neighbor interactions accounting for most of the variation of in f, with concentration appears (28) in some ways to be a basically more satisfactory model for the thermodynamic problem. Thus In f, is then linear not in C'/' but in C1/' over an appreciable range of concentrations C. Already in 1918 (261, Bjerrum had pointed out that the freezing-pointdepression data for a number of salts were consistent with a linear dependence of the osmotic coefficient on C1/%nd had suggested a similar cube-root relation for activity coefficients that could be extended to quite high concentrations by allowance for hydration effects (cf. 31). However, for very dilute solutions, the limiting law must hold. By discussion of such a model, the connections between the behavior of ions in solution, on the one hand, and ions in crystal lattices or in molten salts (an important field in electrolyte electrochemistry), on the other, can be made more clear to the student. Thus, from a modern point of view, it may be desirable to introduce a more criticd attitude to the Debye Huckel model rather than to present the theory with the various degrees of approximation and extensions which are usually given in current and earlier textbooks. The deviations in C from the square root law and also from
the quasi-lattice cube-root law, which occur at high concentrations, must, of course, still be discussed in terms of mutual salting-out amongst the ions (an early idea pro posed by Butler (30) in 1929) or more directly in terms of hydration (26, 29). Such an approach allows for the fact that the electrolyte ions find themselves, with respect to the concentration of typical bulk solvent, at an effectively higher concentration (or activity) then corresponds to their stoichiometric concentration with respect to total solvent moles present, when the amount of solvent held at a "lower activity" in their solvation shells becomes comparable with that available in the whole solution. This usually begins to occur significantly between 1-5 M depending on the size and valence of the ions. In examining the applicability of a cuberoot law for activity coefficients and conductance, i t is, of course, also necessary (31) to correct for solvation effects (26) at the higher concentrations in order to evaluate correctly the nearest-neighbor ionic interaction effectwhich depends on the time-average interionic separation, from which the cube-root devendence on C originates. In a modern presentation of the subject of ionic solutions, i t also appears desirable to stress much more strongly the importance of ionic solvation rather than to introduce this as a "catch-all" for deviations from the Debye-Huckel law a t moderate and high salt concentrations. Both the concepts of solvation energy and primary solvation numbers need to be introduced at an early stage if a real appreciation of the factors determining (a) salt solubility, (b) ionization of acids and bases, (c) dissolution of metals, and (d) standard electrochemical reversible potentials is to be gained, not to mention the important role of solvation energy, entropy, and volume of the activated complex in relation to corresponding quantities for the initial state in solvolytic reactions and inorganic ionic redox reactions (52,SS). A discussion of such factors can also usefully serve to bring out the relation between the individual properties of ions in solution and the dielectric properties of the solvent, e.g., with regard to discreteness of the solvent. its chemical association or otherwise, and the dependence of its dielectric constant on field intensity near ions (34). Most of these basic electrochemical matters are rarely mentioned even in recent general textbooks on physical chemistry for the undergraduate. This approach also serves to provide an opportunity for discussion of individual properties of ions in relation to their effects on one another and on the local structure and polarization of the solvent. Thus, recent work with mixed electrolytes indicates that a generalized use of "ionic strength" as a parameter irrespective of the type of ions concerned is probably unrealistic except a t very low concentrations. Interionic effects are hence to a significant extent specific, as well as the ion-solvent interaction effects which lead in art to the former soecificity. The latter type of short-range interactions can only be examined by non-thermodynamic derivation of the individual ionic properties from the measured thermodynamic properties for the pair (or more) of ions forming the electrolyte; such properties are the partial molal volume (24) or compressibility (24, 46) of the salt a t infinite dilution, or the heat of solvation of the salt a t infinite dilution (25, 88). Here it is important for Volume 44, Number 10, October 1967
/
557
students to recognize that although thermodynamic limitations of principle prevent the measurement of iridividual thermodynamic ionic properties (since no experiment can be devised to measure the chemical potelltial of a single ionic species in solution), non-thermodynanlic analyses (usually based on some theoretical principle) can go a long way (e.g., see ($5))toward a satisfactory semiquantitative evaluation of certain individual ionic propert,iessuch as the heat and standard entropy of solvat~ioiiof ions and individual partial molal volumes. By contrast, such limitations do not apply to certain dynamic properties such as ionic mobility which can, of course, be evaluated unambiguously for individual ions at finite and limitingly zero ~oncent~rations by experimental measurements of transference numbers. The importance of evaluation of individual ionic properties in solution is not to he underemphasized; the geuerality of the concept of "ionic strength," developed by G. N. Lewis and employed by Debye and Hiickel, tended to diminish the importance of the chemical nature of salts in dilute solutions. However, a t moderate and high ionic strengths, ionic solutions exhibit a surprising degree of specificity in their behavior for reasons connected with hydration effects. In biological systems, this specificity of ionic properties is of great importance. Such effects are probably in part the basis of differential permeability of cell membranes, e.g., in the nerve cell, where ~oncent~ration potentials are set up and discharged in the "action cycle" of the neurone. Similarly, in osmotic regulation mechanisms, the ion association arid hydration of protein molecules in varying conformations are believed to be of importance. Finally, it may he noted that incomplete dissociation has ret,urned as a subject of current importance and the nat,ure and thermodynamics of ion-pair formation is receiving much attention. The demise of the Arrhenius theory of incomplete dissociation, which was brought about by the proper realization of the significance of the concentration-dependence of equivalent conductance and mean activity coefficients, has tended for many years to obscure the significance of the real degree of incomplete dissociation which can he measured for a number of orgauic and simple inorganic salts of higher valeuce types, e.g., R4gS04; similar effects determine to an important degree the behavior of linear polyeleetrolytes iu solut,imr, e.g., where anomalous t,ransport of gegen ions with t,hepolyiou can hedirectly determined, e.g., by use of radio-labelled ions (55). Incompletely dissociat,ed electrolytes, besides exhibiting anomalously high Onsager conductance slopes (-dA/d C1'*),are associated with strong dissociation field effects (second Wien effect) arising from field dependence of their association equilibrium const,ants. An important aspect of the behavior of electrolyte solut~ions,that brings the subject back into closer relat i m with problems concerned with non-electrolyte solutions, is t,hc role of non-electrostatic effects. For many years, sncli effects tended to be neglected in theories of electrolyt,e solut,ions despite the fact that both general and more det,ailed theories have existed for a long time for non-electrolyte solutions, e.g., the work of van Laar, Scatchard, Hildebrand, and others; also, the role of the relative size factor in binary solutions was already formulat,ed in 1937 by Fowler and Rushhrooke (36) in relation to non-ideal ent.ropies of mixing in t,wo component 558 / Journal of Chemical Education
solutions, but was only applied to electrolyte solutions quite recently (57). The neglect of such factors for a number of years was, it may be supposed, a result of the surprisingly general applicability of t,he Debye-Hiickel theory for dilute solutions of electrolytes in respect to such thermodynamic quantities as activity coefficients, partial molal heat content.^, entropies, compressibilities, and volumes of salts in dilute solution. Only with the growing appreciation of t,he special nature and properties of the water solvent (38), and the developing interest in strong and very strong solutions, have anomalous and specific nonelectrostatic effects in ionic solutions become recognized as a characteristic feature of aqueous electrolyte solutions particnlarly a t moderate and high concentrations of salts, especially those comprised of a t least one type of organic ion. This aspect of the subject has been intensively developed by Frank and co-workers (59) following earlier X-ray diffraction work on water and solutions by Stewart (40) and by Morgan and Warren (4i),infrared studies by Suhrmann and Breyer (@), and Raman studies by Cross, Burnham, and Leighton (43), with more recent critical work by Walrafan (44) and Falk (45). The short-range ordering and disordering effects which ions have in liquid water ("structure-making," ',structure-breaking," "iceberg" formation effects) appear to be of great importance in many ionization phenomena in solution including (46-48) the thermodynamics of ionization of acids and bases. Such effects arenow regarded (49) as potentially of great importance in the operation of biological processes a t the molecular level, e.g., folding and unfolding of proteins in living cells, behavior of membranes in cells, and the action of various drugs on cellular and enzyme activity; indeed, one may speculate that some of the most important applications of theories of electrolyte (including organic electrolyte) solutions will be in this field of molecular biochemistry in t,he next 20 years. Double-Layer Phenomena
A useful area of contact between solution electrochemistry and the subject of electrode processes which can profitably be developed in the teaching of these topics is the quest,ion of the double-layer. Thus, concepts and treatments of the double-layer, on the one hand, are basic to the discussion of electrode processes, and on the other, depend very much on knowledge of specific properties of ions in solution, e.g., their hydr% tion energies and the radii of primary hydration (coordination) spheres of such ions. Additionally, there are close connections wit,h the Debye-Hiickel theory (see below). Introduction of an elementary treatment of the double-layer in electrochemistry courses also provides an import,ant link with colloid chemistry since the modern theory of colloid stability, the theory of transport of colloidal particles, and the role of colloid chemistry in the hio-sciences all depend very much on the theory of the double-layer a t colloid particle interfaces. Thus, important aspects of the subject of enzyme-substrate and enzyme-inhibitor interactions are bound up with the constitution of the double-layer a t enzyme interfaces in relation to the electrochemical ionization of the enzyme interface and the resulting hydration of ionic centers. Historically, the treat,ment of the double-layer can be
considered in relation to the two extreme views of Helmholtz (fixed condenser-like double-layer of charges, constant field in between) and of Gouy (distributed pointcharge region with continuous variation of electric potential and field up to the charged interface itself). As we have mentioned above, it is usually rarely recognized in treatments of the thermodynamics of ionic solutions that the theory of the double-layer in terms of a continuous distribution of charges in the field set up by the excess charge on a plane interface was treated in 1913 by Chapman usingprinciplesvery similar to thoseemployed in 1923 by Debye and Huckel for the calculation of the distribution of counter-ions about a given reference ion, and the resulting average potential due to the atmosphere of like, andgegen, ionsabout agiven reference ion. Thus, the emential feature of employing a Boltzmann distribution function exp [- z,e$,/kT] fnr each type of ion in conjunction with Poissou's equation
for potential $, a t distance r from the interface iu the double-layer problem, in relation to the space-charge density p, near the interface, was employed already by Chapman and used later in a similar form by Debye and Huckel. The only difference in the initial treatments of these two problems lies in the difference iu geometry of the distribution of charges and t,he resulting potential varies normally to the interface while at mi ion it varies radially, and the distribution of charge is spherically symmetric a t an ion not suffering any net displacement (Brownian motions are, of course, still occurring). An important developmeut in double-layer theory was made in 1924 by Steru (50) who wmbincd au aspect of the "fixed layer" theory of Helmholtz with the idea of a distribution of ions (inherent ill the Gouy-Chapman treatment) determined by the opposing effects of ion interaction (through the zie+ term in the Boltzmann distribution funct,ion) and thermal disordering (through the kT term). In effect, Steru introduccd simultano ously two principal ideas, essential iu the modern theory of the doublolayer: one, the recoguitiou of a distance of closest approach of ions of finite radii to the plane of the interface; and two, the recognition that for various reasons ions may be "specifically adsorbed" (i.e., chemisorbed) a t the interface. The lat,ter effect is related to that implicitly involved in Helmholtz's picture of the double-layer, though the pntential profile (a sharp fall of potential) very near to the interface cau hc sufficiently accounted for by the intrr~ductinunf the recognition that real ions have finite, and not zero, radii. The charge distribution then is cut off at a certain critical average distance from the interface with the result that the potential profile is a discontinuous m e as shown in Figure 1. Here it is interesting and useful for the student to recall that in the Debye-Hiickel theory, a closely analogous cuboff effect is introduced in t,hetreatment for other than very low concentrations by regarding the ions as having a characteristic "distance of clnsest approach," a,determined by their crystallngraphic radii and effective thicknesses of their solvatiou shells when in close juxtaposition. In the Debye-Hiickcl theory, the total net charge residing in t,he ionic atmosphcre, equal and opposite to that (zie) of the central reference ion, is given by
Figure 1.
Potential profile in the double-layer according to Stern's model.
where p,, the space cbargc density at r, is determined by aBoltzmanu distribution for cations aud anio~isthrough the electric potential +,; $, can then be obtaincd after integration. Similarly, for the double-layer pn~blein, t.he uet charge, q,, residing per cmz in the distrihutiou of charges (ill solution) conjugate to the net charge on the interface, qm, is given by application of a form of Gauss's relation
where a is agairi a cut-off dist,ance of closeat approach, this time relevant to the approach of solvated ions to the interface, charged to the extent q, per em2. t'or symmetrical elect,rolytesof valence z, a solution for the field a+,/& obtained by integrating Poisson's equation enables @+,/&), to be written, so that (51, $$),
+.
+,
where is the potential at the cut-off limit a, is a reference potential in the solution at r - m , C is theionic concentration, and a is an average dielectric constant (bearing in mind that the dielectric constant of the solVolume 44, Number 10, October 1967
/
559
vent may vary with field (34), and hence witahr across the doublelayer). I n making a comparison between the Poisson-Boltzmann theory of the double-layer and the Debye-Huckel theory, it is of interest to not,e t,hat no approximations are made in the former theory in regard to linearization theory for electric potential J.,. I n this regard, it may be noted that for symmetrical electrolytes the expression for the space charge density, from the Boltzmann distrihut,ion,is of t,he form and withz+
=
lz-I (= z ) ;
V+
= v-
(=
u);
n+ = n-
very complete knowledge of fact,ors determining electrochemical adsorption of ions and neutral molecules at charged interfaces. Basically the surface tension y of a metal electrode interface depends on (a) the electric surface charge density q;, owing to repulsion effects between like charges in the interface. The relation due to Lippmann is
for the variation of y with electrode potential E a t constant temperature and composition (n,, n,, etc.) of the solution; (b) in the presence of adsorbahle materials at various chemical potentials p,, t,hc Gibbs adsorption equation gives
(= n) 0, =
nzu(exp [-ze+JkTI
- esp
[ee+,/rlTl)
This can be expressed in the form which, it will be seen, has the same form as the expression for the charge density in the doublelayer theory except that J.,is now a function of radial distance from the charge-bearing center. I n the double-layer case, J.,can be more easily solved than in the spherically symmetric case of an ion in solution, where linearization of exp terms (in the sinh) is necessary in order to obtain a convenient solution. The difficulty in the latter case arises since the vZJ.term for the case of spherical symmetry has theform l/r2(b/br) (r2bJ./br) whereas V2J.forthe doublelayer case is simply b2J./brZwhich is readily converted t.o a first order differential by use of the ident.it,y
Solution of the resulting Poisson-Boltzmann differential equation for the double-layer t,hen leads to the result shown above without the necessity of linearization of exp terms. In both the Debye-Huckel and the double-layer theories, however, use of the Poisson equation implies that the actual distribution of individual ions can be replaced by an equivalent smeared-out continuous charge distribution giving the volume charge density a t any point. The real discreteness of the charges can be important in the theory of the double-layer (52) since, twodimensionally, relatively high ionic concentratioris are involved near the electrode surface; similarly, discreteness-of-charge effects tend to invalidate (28) the ionic atmosphere model for electr~lyt~e solutions at concentrations higher than 0.14.5J6,as we have mentioned above. Under these conditions (quasi-lattice situation), the non-ideal free energy contribution (- RT In f) then depends on the cube mot of concentration, instead of on the square root as given by the familiar DebyeHuckel relation. Similarly a nearest neighbor interaction theory for a two-dimensional lattice of charges in the double-layer gives a square root dependence on coverage for the free energy of adsorption of ionic species. Most information on the double-layer in electrode processes is derived from elect,rocapillary (surface tension measurements) and capacity measurements on liquid metal electrodes, especially mercury. Here the classical equations of Lippmann and Gibbs lead to a
-
560 / Journal o f Chemical Education
where r, is the "surface excess" of adsorbate j related to the surface coverage of the electrode by j . The interface capacity C can be expressed differentially from Lippmann's equation as
and can also he measured directly by ac or dc charging methods. 'Very complete information can thus be obtained by electrocapillary and capacity measurements, including amounts of substances adsorbed (including contributions from each type of ion in solution), charge in various regions of the solution side of the doublelayer, charge on the metal surface as a function of potent,ial E (referred to some reference electrode), and, by use of eqn. (6) and other relations, the evaluation of J.,. Much recent work (53) has been concerned with distinction between specific chemisorption and normal electrostatic effects such as have been discussed above. In particular, adsorption isotherms governing the interactions of ions and molecules in the interfacial layer, both among themselves and with t.he surface, have been evaluated in considerable detail. Such information is of great importance in the interpretation of electrode processes involving such ions and molecules (see below), and electrical effects in the thermodynamics of adsorption can be evaluated in addition to the chemical factors normally involved in adsorption at surfaces. In general, a t met,al electrode interfaces, considerably more quantitative information can be obtained about the double-layer than in the case of colloid interfaces, and in the pedagogical treatmcnt of the latter subject from the point of view of colloid chemistry, there is much to be said for introduction of that topic from an electrochemical direction, rather than from the classical direction of "hydrophobic sols," "electrnphoresis," "zeta potentials" (the analog of the potential discussed above), and similar matters. Electrode Reaction Kinetics and Electrocatalysis
A great body of the early and now classical work on electrochemical reactions was concerned with the thermodynamics of reversible cells according to the principles of Nernst, and indeed this approach has led to very precise data on a variety of reactions involving ions. Data derived by such classical studies were used largely in two ways: (a) to provide in part the basis of the Third Law of Thermodynamics; and (b) to provide ac-
curate activity coefficients at low and intermediate concentrations which enabled the Debye-Hiickel theory to be tested with a high degree of rigor. While such electrochemical studies were thus of prime importance in the development of thermodynamics and the theory of electrolyte solutions, they tended to withdraw interest from the kinetic and mechanistic aspects of electrode processes. Kinetic aspects of electrochemistry-which constitute some of the most important and the most active areas of the subject a t the present t i m e a r e curiously enough often presented in textbooks (including quite recent ones) from a remarkably out-of-date point of view. Thus, emphasis is often given to "decomposition potentials" (an arbitrary concept) in relation to reversible potentials of the electrodes comprising a cell, and the involvement of the so-called "overpotential" associated with the passage of net electrolysis current in a cell is regarded almost as an "unpredictable result" associated with that passage of current. While the essential theory of the kinetics of electrode reactions appeared correctly (following earlier original publications) in an important monograph "The Theory of Rate Processes" already in 1941, many subsequent textbooks, including some in which kinetics of other types of processes are dealt with from a modern point of view, have failed to present kinetics of electrode processes in a comparable context with treatments for regular homogeneous and heterogeneous processes. In fact, when presented in the proper manner, kinetics of electrode processes can be used to illustrate and complement the treatment of other heterogeneous processes in a useful way. In a modern presentation of the subject for undergraduates, it is desirable to develop the subject along the following lines: In most non-electrochemical reactions, the processes of relative electron donation and acceptance cannot he distinguished, and we can only say that in the "product" of a reaction there has been some redistribution of electronic charge (and of the atomic nuclei) compared with the situation in the "reactant" particles. In an electrochemical process at a given electrode, we are able to study separately each side of the "oxidation-reduction" process, which, in a general way, characterizes all chemical reactions; thus, at a cathode, the process of electron acceptance can be examined, and at an anode that of electron withdrawal. Since the reaction of electron transfer in electrolysis occurs at an interface, the reaction is heterogeneous, and this fact therefore necessitates the consideration of all the factors involved in adsorption and catalysis at metd interfaces; such factors will be complicated in the case of electrode processes by the ubiquitous presence of solvent and by the profile of electric potential at the interface. Since electron transfer to a reactant ion, e.g., H30+, usually results in the formation of a free-radical (except, however, in ionic redox reactions, e.g., Fe3+/FeZ+; Cr3+/Crz+),surface mtalysed reactions between such free radicals, following the electron transfer, are an important aspect of most electrolytic reactions. In some cases, e.g., the electrochemical oxidation of hydrocarbons. methanol. formic acid. and hvdrazine. surface catalyse'd dissociative chemiso&tion processes bccur on the electrode prior to electron transfer steps which then
usually involve the dissociated fragments of the initial reactant. This is the basis of studies in the field of " electrocatalysis." The electrode metal surface, acting as a catalyst, causes breakdown of the adsorbed snbstrate molecule and successive electron and proton transfer events then complete the oxidation of the fragments often (formally) to form HzO and COz. The types of electrochemical steps that occur depend on the coverage of the electrode by the fragments and the cocoverage by species electrochemically formed from the solvent, e.g., H and OH species in the case of aqueous solutions. Since electrode reactions are heterogeneous, the relevant rate equation must include surface concentration terms, and these must usually be expressed in terms of bulk concentrations through an isotherm determined (e.g., if the electrode is Hg) by means of capacity and/ or electrocapillary measurements, or (if the metal is a solid one) by other means such as the use of reactants (54) labelled by radio tracer elements such as 14C(p-) or 15S(p-), by ultraviolet spectrophotometry (56), or again by capacitance methods. Finally, the most important factor which formally distinguishes electrode kinetics from kinetics of other heterogeneous processes is the fact that the rate constant of the reaction is potential dependent. In the same way as the rate constant of a non-electrochemical reaction is written as k
=
k?'
exp [-AG'~/RT] h
(10)
for a standard free energy of activation &Go', the rate constant K for an electrochemical charge transfer step may he written as
where AGOr is formally an electrochemical standard free energy of activation and is dependent on a fraction p of the electric potential difference A V operating between the metal and the layer of adsorbed reactant, k
kT
= h exp -
[AG'f - zF@AV]/RT
(12)
for a charge transfer event involving z electrons; usually z = 1 in most electrochemical kinetic events and p is approximately 0.5, since the transition state may be regarded as usually arising half way between the reactaut and the metal surface, across which region
Figure 2. Potential energy diagrams showing the relation between the. symmetry coefficient @ in electrochemical kinetics and the a coefficient in Bronted'r relation in connection with changer in free energy of octiv&ion.
Volume 44, Number 10, October 1967
/
561
the potential AV operates. More sophisticated views of the significance of B relate this quantity either to the relative slopes of potential energy curves for reactant and product molecules, or ad-species at the crossing regiou (transition state), or to the average charge on the activated complex. The former representation is easily depicted as shown in Figure 2a where it is apparent how the activation energy for charge transfer can become changed by variation of potential difference at the electrode interface; this situation is a very useful one for introducing the elements of absolute reaction rate t,heory in the teaching of kinetics in general, and for illust,rating quantnm-mechanical principles in electrochemical kinetics. In regard t.o the latter representation, it is possible to make an analogy to the effect of changes of pressure on a reaction rate constant. With the definition of k in terms of AGOf given above it is apparent. that
I = RT' ( A 4
where AV' is the change of volume in the activation process as the reactants approach the transition state. The nnalogous expression in elertrode kinetics is since the AGO%term is, it is supposed, independent of AT7. The quantity b e , which we have represented as aqt, is seen to be equivalent to the change of effective charge in formation of the activated complex, a result consistent with the alternative formulation of p in terms of average charge on the activated complex discussed by Hush (56) and mentioned above. In relat,ion to double-layer theory, the potential AV is the difference between V, the total metal-solution potential drop, and the potential in the double-layer at the cub-off distance a or at some plane closer to the surfare when specific adsorption of reactant ions is involved. Where possible, it is useful in the presentation of electrode kinetics to illustrate connect~ionsof the subject with other general and more familiar aspects of physical chemistry. An interesting example is concerned with the relation between the form of the equation for ii (eqn. (12)) and Br@nstedJsrelation for rates of protonation in a aeries of acid-base reactions as a function of changes of t,he acid-or the conjugate base-strength through the equilibrium constant. The representation for Br@nst'ed's or coefficient shown in Figure 2 is seen to be analogous (cf. (67))to the representation as shown in Figure 2 for the electrode reaction case. In the latter case, changes of AV modify AGO^ in a way similar to that whereby changes of acid or base strength in acid-base equilibria modify the proton transfer rate constants. In fact, in the discharge step of the hydrogenevolutionreaction
+.
M
+ H%O++ ew,
-
+
M H~~d~,>~ba*, I110
1
Hz the metal cathode can be regarded in the general case as a Lewis base of vaqing base strength depending on the elertron charge density q, (positive or negative) on the interface. At a metal such as Hg, q, is susceptible to direct measurement and the excess charge can be varied between zt0.15 electrons/atom in the surface layer.
-
562 / Journal o f Chemical Education
The practical result of the exponential dependence of i;: on AV is that the rates of charge transfer steps in electrode reactions, and hence the current densities, are logarithmic functions of the potential V or AV, and the slopes of such so-called Tafel relations can, in favorable cases, be characteristic of the reaction mechanism. This result arises simply from the application of an Arrhenius type of relation (58) with a potential-dependent activation energy. The form of the equation for enables it to be seen that the applied potential V, and, hence, the related overpotential (defined by the difference V - V,, where Va is the metal-solution p.d. at the interface when the reaction concerned is a t electrochemical equilibrium and the net current is hence zero) are fundamental quantities in the kinetics of charge transfer and that the rates of all charge transfer processes will be susceptible to a potential dependence of this form. The commonly made classification of electrode reactions as "reversible" and "irreversible" is thus rather misleading and certainly arbitrary. The only practical difference between one group and another lies in the value of the so-called exchange current, i.e., the magnitude of the equal and opposite currents, io, (related to the values of ic (eqn. (12)) when V = Ve for forward and reverse directions of the reaction) passing per cm2 when the reaction is a t electrochemical equilibrium. Such&values canvary from 10-13amp cm-%to 10W3amp cm-%for the hydrogen evolution reaction at Hg and Pt, respectively, and can become as large as 1-10 amp ~ m for - ~some metal deposition and redox reactions; in these cases, the reactions are usually diffusion controlled. Some discussion of polarography (and electroanalpis) may serve as a useful bridge between the subject of reversible electrode potentials and electrode kinetics. After recognition of the central role of electric potential in the discussion of kinetics of electrode reactions has been made, it becomes easier to present to the student problems concerning rates of electrode processes if the kinetics are treated a t constant potential; thus, under such a condition, the kinetics may be examined, e.g., with regard to reaction orders, surface coverage by adsorbed intermediates, etc., in the same way as would a non-electrochemical heterogeneous reaction. Experimentally, this situation may be easily realized by use of a so-called "potentiostat" which automatically regulates by a feedback circuit the current passing a t an electrode so as to maintain the potential of the electrode a t some fixed value with respect to the potential of a reference electrode, this value being set by a prior adjustment of some internal reference voltage in the instrument itself. The direct and simple dependence of electrochemical rates on potential enables an electrode nnder control by means of a potentiostat to be programmed with respect to potential by feeding into the reference channel of the potentiostat a timedependent potential having, e.g., a linear form or a step form. The electrode reaction then undergoes a kinetic relaxation in response to the applied "potentiodynamic" signal. With a step signal, this method allows the determination of rate constants as in other step relaxation techniques (e.,g. temperature jump, or pressure step methods) in kinetics. In the case of an electrode a t which an adsorbed intermediate (e.g., H in the hydrogen evolution reaction) is being deposited or removed over a given potential
range, the time integral of the 'urrel~t passiug gives the charee associated with this de~osition or removal process and consequently the change of roverage by the intermediate. This method, following a controlled potential-step pretreatment program, is the basis of much of the recent work on the nature of intermediates adsorbed at the electrode interface in electrochemical catalytic reactions. I n cases where the intermediate itself may not suffer any direct charge transfer oxidation or reduction, the direct determination of chauges of accommodation at the surface for electrorhemically adsorbed H atoms can provide a useful in situ estimate of the coverage by the "electro-inactive" intermediate. I n this respect, adsorption a t electrode-catalyst materials can he studied in at least as much detail as at ordinary heterogeneous catalysts. The electrochemical study of adsorption of intermediates by a voltage sweep procedure is in fact closely analogous to the method of "flash desorption" which is now widely used for detection and analysis of species adsorbed at ordinary catalyst surfaces where-a programmed elevation of temperature is employed. Similarly, elcctrolysis under a square-wave periodic program is analogous to photochemical studies with a rotating sector technique (59). Electrode surfaces under potential control can be regarded as catalyst surfaces having potentially variable and controllable adsorptive and Lewis acid-base character. Again this aspect of electrode processes forms a useful basis on which the student can be made aware of the relations of the subject to other important branches of physical chemistry.
-
Literature Cited
( 1 ) DEBYE,P., and HUCKEL,E., Z'hys. Zeil., 24, 49, 185, 305 (19231: , , , 25. 97 (1924). , ( 2 ) ON~AGER, L., Phys. Zeil., 27, 388 (l!l"6); 28, 277 (1927). W., AND CARLISLE,S., Nicholson's J . ( J . Nat. ( 3 ) NICHOLBON, Phzl.) 4 , 179 (1800). M., "Experimental Researches in Electricity," ( 4 ) FARADAY, Series 111, 1833; [Phzl. Trans., 123, 23 (1833); 124, 77
. .
,.""=,,.
llO?d~l
( 5 ) BERZELIUS, J. J., AND HISINGEI~, M., Ann. de Chim., 51, 172(1804); Gilb. Annalen, 27, 270 (1807). ( 6 ) DANIEL,J. F., Phil. T ~ a n sRoy. . Soc., London, 129,97 (1839). ( 7 ) OSTWALD, W., Zeit. phys. Chem., 2,36 (1888). S., Zeit. phys. Chem., 1, 6'31 (1887); see also ( 8 ) ARRHENIUS, "Theories of Solutions," Yale University Press, New Haven, Conn., 1912. F., Ann. Physik, 26, 108 (1855). ( 9 ) KOHLRAUSCE, ( 1 0 ) BJERRUM,N., Zeil. Elektrochem., 24, 321 (1918); Zeit. Anorg. Chem., 109, 275 (1920). ( 1 1 ) MILNER.S. R.. Phil. Mao. 23. 55 (19121: 25. ~,742 11913). (nj CHAPMAN,D. L.,Phil. ~ a q25,4?s . (~Sis). ( 1 3 ) GOUY,G., Ann. Chim. Phys., ( 7 ) 29, 145 (190'3); ( 8 ) 8 , 291 (1906); J . Phys., 9 , 457 (1960); Ann. Phys. (Paris),
(24) MUKEETEE, P., J . Phus. Chem., 65,740 (1961) AND PASSYN-
.
sm. A. G.. Aeta Phssieoehim.. U.R.S.S.., 8. 385 11938). . . HALLIWELL, H. F., .ANDNYBURG,S. C., Trans. Faraday Soe., 59, 1126 (1963). BJERRUM, N., Medd. vetensk Akad. Nobelinsf, 5 (No. 16) (1919); Zed. Anorg. Chem., 109, 275 (1920). GHOSH,J. C., 3. Chem. Soe., 113, 449 (1918). FRANK, IT. S., A N D THOMPSON, P. T., "Strncture of Electrolytic Solutions" (Editor: W. HAMER)John Wiley $ Sons, N. Y., 1959, Chap. 8 ; see also KORTUM, G., ''Tseatise on Electrochemistry," (2nd ed.), Elsevier Press, New York, 1965, p. 20. STOKES,R.. H., AND ROBINSON, R. A,, "Eleclmlyte Sdutions," Butterworths, London, 1956. BUTLER,J. A. V., J . Phy6. Chem., 33, 1015 (1929). DEsNoYEns, J. E., A N D CONWAY, B. E., J . P h y ~Chem., 68, 2305 (1964). JR., A., "Solvolytic Displacenlenl e.g., see STREITWIESEIL, Reactions." McGraw-Hill. N. Y.. 1962. ( 3 3 ) LAIDLER,K. J., A N D SACHER,N., "Modem Aspects of Eleotmchemistry," (Edifors: J. O'M. BOCKRISAND B. 13. CONWAY) Butterworths, London, 1964. Chnp. 1 of Vnl T T T
( 3 4 ) BOOTH,F., J . Chem. I'hys., 19,391; 1327; 1615 (1951).; see abo ('lRiutA~E, D. C., J . Chem. Phys. 18, 903 (1950). R. H., J . Amer. Chena. Soc., ( 3 5 ) WALL,F. T., A N D DOREMUS, 76. 1168 (1954): , ., see also WALL.F. T.. 4ND HILL.W. B.. .T. Amer. Chem. Soe.. 82. 5599 (19fi01 A N D WALL.F. T..
.- , - - - , ---- - .. ...,. ( 3 6 ) FOWLER, R.. IT., A N D RUSHBRO~KE, G . S., Trans. F a r d a y Sac., 33, 1272 (10371. ( 3 7 ) CONWAY, B. E., A N D VErmnLL, R. E., J . Ph@. Chen~.,7 0 , E., l'ran~.Fa~arlau 1473 (1966): see also GLUECKAUF, \
,
~
.
-.,.
7 , 1R2 11417) \ A""
&"
, ,~
,
". .
5 ( I ! ) NEMETHY, G., A N D SCHEHIOA,11. A,, 1. Chem. Phys., 36, 3382; 3401 (1962). ( 3 0 ) FRANK,13. S., AND EVANS,N., J . Chem. Phys.. 13, 507 (1945); see also FRANK,H. S., and WEN, W. Y., Discussions of the Famday Soc., 24, 133 (1957). (40) S ~ ~ A R G.TW., , Phys. Rev., 37.9 (1931). ( 4 1 ) MORGAN, J., AND WARREN, B. E., J . Chem. P h y ~ . ,6 , 666 (19310. , , ( 4 2 ) SUHRMANN, n., AND BEEYER,F., Zeil. phys Chem., BZO, 17 (1932); B23,193 (1933). ( 4 3 ) Cnoss, P. C., BURNHAM, J., AND LEIGETON, P.A., J. Amer. Chem. Soc., 59, 1134 (1937). ( 4 4 ) WALRAFAN, G. E., J . Chem. Phys. 36, 1035 (196"); 40, 32411 (1964). (45) FALK,M., A N D FORD,T. A,, Can. J . Chew., 44, 1699 (1966). B. E.. AND VERRALL. R. E.. J . Phm. Chem.. 70. ( 4 6 ) CONWAY. ( 4 7 ) WEN, W. Y., AND SATL'D, S., J . Phys. Chem., 68, 2639 (1966); HEPLER,L., J . Phw. Chem., 69, 965 (1965). ( 4 8 ) CONWAY, B. E., Ann. Rev. Phys. Chem., 17,481 (1966). ( 4 9 ) SCHERAGA, 11. A., Ann. N . Y . .id. Sci.. 125, (2), 249 (1965). ( 5 0 ) STERN,0.) Zeit. Eleklrochem., 30, 508 (1924). It.. "Modem A s ~ e c t s of Electrocherniat.rv." (,5 1,) PAR~ONS. ", (Edilor: J . OW. BOCKRI~) Butterworths, London, 1954, Vol. I, Chap. 11. ( 5 2 ) GRAHAME, 1). C., Zeil. Elektroehen~., 59, 773 (1955); see a130 LEVINE,S., BELL, G. M., A N D CALYERT, I)., Can. J . Ch,em., 40, 518 (1962). 1). C., e.g., see Chem. Rev., 41, 441 (1947) and (53) G~UHAME, later DaDers bv this author: see also PARSON& . R.,. l'rans. Farada!;~oc.,'51,1518 (1955). ( 5 4 ) WROBLOWA, TI., ,AND GREEN,M., Eiectmchin~.Aeta, 8 , 679 (1963l. ~- - - , (55) CONWAY, B. E., AND BARRADAS, R. G., J . Ekctlwanal. Chem., 6 , 314 (1063). ( 5 6 ) HUSH,N. S., J . Chem. Phys., 28, 962 (1958). ( 5 7 ) FnuMKrN, A. N., %it. phw. Chem., 160,116 (1932). J. O'M., "Electrical Phenomena at Interfaces," ( 5 8 ) BOCKRI~, (Editor: J. A. V. BUTLER).Methuen. London. (1951). chap. V I I ; see also BOCK& J . o'M.,Chem. R& 43; 525 (194R). .-. ~. -.,. ( 5 9 ) WILSON,C. L., AND LIPPINCOTT, W. T., J . A n m . Chem Soc., 78, 4291 (1956).
~ - ~ - ~ .
( 1 4 ) VAN'T HOPF,J. H., Zeit. phys. Chem., 1,481 (1887). ( 1 5 ) TAFEL,J., Zeit. phys. Chem., 50,641 (1905); 34,187 (1900). ( 1 6 ) GURNEY, R. W., P ~ cRoy. . SOC.,London, A134,137 (1931); A186. 378 (1932). (17) BAAR~,'M., ~itzbe;.Natuwiss. Marburg, 63, 213 (1928). T.,AND VOLMER,M., Z. p h p Chem. (18) VON ERDEY-GRUZ, A150, 203 (1930). ( 1 9 ) FRUMRIN,A. N., Zeit. phys. Chem., A I M , 121 (1933); cf., also Aeta Physieochim., U.R.S.S. 6 , 502 (937). ( 2 0 ) BUTLER,J . A. V . , PTOC.ROU.SOC.. London, A157, 423 (1936). 121) BUTLER,J . A. V., Trans. F a ~ d a ySoc., 19, 734 (1924). B. E., "Theory and Principles of Electrode ( 2 2 ) CONWAY, Processes," Ronald Press, Inc., N. Y., 1965, p. 3. B. E., Ree. Chem. Progvess, ( 2 3 ) BOCKRIS,J . O'M., AND CONWAY, 25,3l (1964).
~
. .
Volume 44, Number 10, October 1967
/
563
