[Comprehensive Chemical Kinetics] Electrode Kinetics: Principles and Methodology Volume 26 || Chapter 2 Mass Transport to Electrodes

Chapter 2

Mass Transport to Electrodes

KEITH B. OLDHAM and CYNTHIA G. ZOSKI

1. Introduction

The investigation of electrode kinetics has one paramount advantage over other kinetic studies: the rate of the electron transfer reaction

Reactants + n e-Products (1) can be measured directly rather than needing to be inferred from concentration changes. This advantage is a consequence of Faraday’s law, which asserts the proportionality of the electron-transfer rate

I Reaction rate = -

nAF

to the faradaic current i divided by the elctrode area A . In eqn. (2), n is the number of electrons and F is Faraday’s constant.

On the other hand, electrode kinetic studies are at a disadvantage compared with investigations of homogeneous kinetics because concentrations are not uniform and surface concentrations can rarely be measured directly (optical methods can sometimes provide direct measure- ment of the product concentration [l]). This means that the converse situation to that in classical homogeneous kinetics exists in electrode kinetics: concentration information needs to be inferred from reaction rates.

To calculate concentrations at the electrode surface requires a knowledge of

(a) the stoichiometry of the electrode reaction; (b) the bulk concentrations of the species involved; (c) the rate of the reaction [or equivalently, because of relationship

(d) the laws governing mass transport for the particular electrode

(e) the prevailing experimental conditions. This chapter is concerned with how one uses items (a)-(e) to calculate

concentrations a t the electrode surface. In the electrocheinical literature, expressions for surface concentrations are seldom regarded as the end result of a transport prediction. Instead, one usually assumes that the surface concentrations of the species involved in the electrode reaction

(2), the faradaic current] since the onset of the experiment;

geometry; and

References p p . 141-143

80

obey either a thermodynamic relation (nernstian conditions) or a particular kinetic expression (volmerian conditions), so that the result of the analysis of mass transport can be presented as a relationship between the experimentally observable variables: current, cell potential, and time. In this chapter, we shall primarily report relationships involving concentrations, since these are the kinetically significant variables.

1.1 SPECIES INVOLVED IN TRANSPORT

Throughout this chapter, we shall specifically exclude electrode reactions that consume or generate insoluble species. Thus, the most general electrode reaction is

v A A(so1n) + v B B(so1n). . . If: n e- v z Z(so1n) + v y Y(so1n) +. . . (3)

where the vs are stiochiometric coefficients, A, B. . . . are reactant species and Z, Y. . . are product species. Usually, all the species are dissolved in the electrolyte solution, but an important exception occurs in the reduction of certain metal ions at a mercury electrode to produce an amalgam

M"+(soln) + n e(Hg)-M(ama1) (4) Cases in which one of the reactants or products is the material of the electrode itself, as in

2 Hg(l) - 2 e (Hg)- - - Hg?+ (as)

CuCl:-(aq) + 2 e (CU) -CU(S) + 4 Cl-(aq)

(5)

(6)

or

are not excluded. There is, of course, always an ample supply of the electrolytic solvent

(often water) and of the electrode material (often a metal) a t the interface, but all other species must be transported to and from the electrode surface as illustrated in Fig. 1.

Usually, there is no significant impediment to the transport of electrons, through this may not be true of some semiconductor electrodes [2-71. When there is more than one reactant, it is usually possible to adjust bulk concentrations (or adopt other experimental strategies such as buffering) so that all reactants except one are in such excess that their transport poses no difficulty. This is an analog of the "isolation technique" familiar to kineticists. The same is true of product species: it is generally possible to arrange experimental conditions so that, at most, only one product species is subject t o a transport restriction.

Hence, we shall customarily ignore all but one reactant species and all but one product species. Moreover, we shall assume that the stoichiometric coefficients of these species are both unity. This is not an essential

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T e lec t rode sur face

Fig. 1. Transport t o and from the electrode surface.

assumption, but it does serve to simplify our arguments and covers the majority of practical examples. Thus, for a reduction experiment, the electrode reaction may be abbreviated to

0 (soln) + n e - R (soln) ( 7 )

where 0 (for oxidized species) is the single reactant species we need consider and R (for reduced species) is the sole product species under consideration. Of course, either 0 or R or both may be ions.

Sometimes, we shall address an even simpler class of electrode reaction in which there is only a single electroactive species of a varible activity. The simplest instance of this class is the reduction of metal ions on a cathode composed of that metal, for example

M"+(soln) + IE e ( M ) - M(s) ( 8 )

This reaction is the only one treated in Sect. 4 of this chapter. It will be our custom to deal with cathodic electrode reactions, i.e.

with reductions like (7) and (8 ) , rather than with the equally important oxidation processes. For this reason, cathodic currents will be treated as positive*.

1.2 THE ELECTRODE SURFACE

The region extending from the phase boundary out to about 3 n m is quite unlike the solution beyond. Generalizations valid elsewhere in the solution do not necessarily apply here. In this inner zone, the so-called double-layer region [9], we may encounter a violation of the electroneutrality condition (see Sect. 4.1) and large electric fields. Concen- trations may be enhanced or depleted compared with the adjacent solution.

* This is the usual convention in electroanaIytica1 chemistry, though it is at variance with the more logical IUPAC convention [ 81.

Referencespp. 141-143

82

Through phenomena inside the double-layer region may have profound effects on the kinetics of electrode reactions [ l o ] , the zone is fortunately of little or no consequence in discussions of transport from the solution to the electrode or vice versa. To appreciate why this is so, consider a typical electrochemical experiment in which species R is being generated at an electrode by a current of density 1.0 A m-2. After 1 s of electrolysis, the transport zone (the region into which R is being carried) will extend into the solution a distance of some 10 pm, in comparison with which the double-layer thickness of 3 nm is negligible: Moreover, each electrogenerated R molecule will transit the double-layer region in less than

When we discuss, in this chapter, concentrations “at the electrode surface”, we will generally ignore the effect of the double layer. Hence, “at the electrode surface’’ really means “outside the double-layer” or, more precisely, “extrapolating to the electrode surface the concentration profile from beyond the double layer”, as illustrated in Fig. 2.

The narrow double-layer zone is the site of all the chemical and physico- chemical processes that attend the electrode reaction. These may include, in addition to the electron transfer itself, adsorption and desorption steps, as well as chemical transformation between 0 and R (which are the species stable in the bulk of the solution) and modified species (with less solvation, perhaps, or with different configurations) o and r which are adsorbable. Thus, the complete train of events may be

O(so1n) R( soln)

Gsoln) e= o(so1n) d o ( a d s ) x r ( a d s ) G=== r(so1n) ===+’k(soln)

or some even more elaborate scheme. Most of these complexities need not concern us in this chapter, but should be noted.

Let ro and rR (with units of mol m-2) denote the amounts of 0 and R (or their modified forms o and r) that are adsorbed. The shaded area in

200 ps.

trans- - 1 port transport double-layer region 1

concentration f,

-distance double layer

Fig. 2. Illustration of electrode “surface concentration”.

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Fig. 2 illustrates the significance of r. The validity of certain equations that we shall develop in Sect. 2.2 requires that Po and rR remain constant. Certain experimental methods, both thermodynamic and kinetic*, can be used to investigate the extent of adsorption on electrodes, but these are beyond the scope of the present discussion.

1.3 ELECTRODE GEOMETRY

We stipulate the electrode to be smooth (though not necessarily flat) and of constant area A . By “smooth” we mean that any undulations in the electrode surface should not exceed the thickness of the double layer. For an electrode that is less smooth than this, the concept of electrode area is somewhat vague and the “effective electrode area” may change with time. By prescribing a constant electrode area, we exclude one of the most practical electrodes: the dropping mercury electrode treated in Chap. 5.

Where it is possible to define the coordinate unambiguously, we shall use x t o denote distance measured normal to the electrode into the transport medium, with x = 0 corresponding to the electrode surface. We may distinguish three simple types of electrode geometry: planar, convex, and concave, as shown in Fig. 3. Transport to a planar electrode occurs along parallel lines. For a convex electrode, transport to the electrode is convergent, whereas transport away from the electrode is divergent. The opposite is true for a concave electrode. The case of a convex mercury electrode at which reaction (4) occurs is unusual but important; there, the transport of both 0 and R occurs convergently.

The region around the electrode, through which transport occurs, is filled with solution and should usually be unimpeded by cell walls, other electrodes, etc. for a sufficient distance. How far is “sufficient” depends upon the mode of transport and on the duration of the experiment. For transport by diffusion alone, the requirements are very modest indeed, as indicated in Table 1.

The electrode at which the reaction under study is proceeding is called the “working electrode”. It is with this one electrode that we are solely concerned. There is, however, an important exception: the experiments

(a) (b) ( C )

Fig. 3. Types of electrode geometry. (a ) Planar; ( b ) convex; (c) concave. The arrows indicate transport lines to the electrode surface.

* Especially chronocoulometry [ 11-14],


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TABLE 1

Unimpeded distances from the electrode surface

Duration of experimentls Unimpeded distancelmm

0.1 0.06 1.0 0.2 10.0 0.6 100.0 2.0

discussed in Sect. 4, in which there are two working electrodes. In those cases, it is important that the two working electrodes be parallel and separated by a rather narrow gap.

1.4 FARADAIC AND NON-FARADAIC CURRENTS

Two distinctly different types of current may flow at the electrode surface. One kind includes those in which electrons are transferred across the interface between the electrode and the solution. This electron transfer causes a chemical reaction, either oxidation or reduction (Sect. 1.1) to occur. These reactions are governed by Faraday’s law, which asserts the proportionality of the electron-transfer rate to the current [eqn. (2)]. Accordingly, this current is called a faradaic current.

A second type of current arises due to the presence of the electrochemical double layer (Sect. 1.2). Additionally, a current may flow due to the adsorption or desorption (Sect. 1.2) or species 0 and R as well as electroinactive species. In these instances, no chemical reaction occurs and consequently electrons are not transferred across the electrode- solution interface. However, a current may flow elsewhere and this current is called a non-faradaic current.

Both faradaic and non-faradaic currents may flow when an electrode reaction occurs. Thus, the total current which flows is often the sum of the faradaic and non-faradaic contributions to the current. Most often, it is the faradaic current that is of interest. Many electrochemical techniques have been developed which minimize or eliminate this non-faradaic contribution to the current, but discussion of these is beyond the scope of the present chapter.

In Sect. 1.2, we chose to ignore the effects of the double layer as well as complexities due to adsorption and desorption processes. Similarly, we will also choose to ignore the presence of non-faradaic currents, though in practice they may be important. Hence, throughout this chapter only faradaic currents will be considered.

1.5 KINETIC STRATEGY

Electrochemical studies are performed for many reasons, but in this chapter our preoccupation is with experiments carried out for the purpose

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potent lo1

initiol electric current profiles conditions

f of 0 ond R of 0 ond R

1 mpose *Experiment Observe 4

‘I J” \ J kinetic relotions tronsport relotions

concentrotion profiles geometr-1 c

conditlons (boundary condition)

Fig. 4. Linkage between kinetics and mass transport. Quantities enclosed by a single frame are (generally) functions of time only, while those enclosed by a double frame are functions of spatial variables as well as time.

interrelotionship of surfoce concentrotions

and surfoce fluxes

geometric conditions predict

lsurfoce fluxes]

Fig. 5 . Indirect method for determining electrode kinetics. In this method, a particular kinetic law has been assumed.

8

of determining the kinetics of the electrode reaction. At first sight, it is not evident how mass transport, i.e. the motion of the electroactive species 0 and R through the solution, interacts mathematically with the elctrode kinetics, which manifests itself only at the interface between the solution and the electrode. The interaction is not direct but occurs via the surface concentrations (see Sect. 1.2) and surface fluxes (see Sect. 2.2), as illustrated in Fig. 4.

The only variables that are generally accessible in the scheme portrayed in Fig. 4 are the current and the potential; in a typical electrochemical experiment, one of these variables is imposed on the cell and the other is observed. How, then, does one learn anything about the kinetics by making use of the known laws of transport? There are two strategies for doing this.

The most common strategy is illustrated in Fig. 5. A potential (constant or varying) is imposed on the cell and the current-time relationship is monitored. In the theoretical segment of the study, one assumes a

concentrotion tronsport


86

interrelationship of surface concentratians

observe current 1 * e impose

y j . t x per i men t

conditions predi c t

/surface fluxes]

Fig. 6. Indirect method in which the electrode reaction is assumed t o be reversible.

current -7

1 mpose observe potent ial ,Experiment

semiintegrate

concentrotlons seek kinetic law

Fig. 7. Direct method for determining electrode kinetics. Contrary t o the indirect method portrayed in Fig. 5, n o particular kinetic law has been assumed.

particular kinetic law (usually volmerian kinetics, discussed in Sect. 3.5) by means of which an equation linking current, surface concentrations, and time is evolved. With the aid of this equation, the transport problem can be solved, generating expressions for the concentration and flux profiles of 0 and R. Then, the fluxes of 0 and R at the electrode surface, and hence the current, can be predicted. Finally, comparison of the measured current with the predicted current is used to verify the adequacy of the kinetic assumption and to provide values of the kinetic parameters.

Commonly, the electrode reaction is assumed to be “reversible”, which simplifies the mathematics considerably because a direct prediction of surface concentrations is possible from the potential alone, as in Fig. 6. However, kinetic information is entirely lacking from experiments conducted under reversible conditions since the electrode reaction is then an equilibrium.

The second strategy which may be used to learn about the kinetics of an electrode reaction is illustrated in Fig. 7. As before, a potential (constant or varying) is imposed on the cell and a current-time relationship is monitored. However, instead of assuming a particular kinetic law, one processes the experimental current by semi-integration (see Sects. 5.2 and 5.4), thus enabling the surface concentrations to be calculated directly. Hence, the kinetics can be elucidated by a study that involves only the

87

electrical variables: potential and current, together with the semi-integral of the current. This is the direct method of elucidating electrode kinetics and is discussed in Sects. 3.5 and 5 . 5 .

1.6 SYMBOLS AND UNITS

The symbols listed below are used throughout this chapter. In most cases, the usage follows the recommendations of the IUPAC Commission on Electrochemistry [ 81 ; however, there are exceptions.

electrode area ( m2) constant of the j t h component (Table 6) constant (Table 6) capacitance (F) capacitance at the input of an operational amplifier (F) capacitance of capacitors j and j - 1 (F) concentration (mol m-3) concentration of electroactive species at the cathode surface (mol m-3) bulk concentration of the jth ionic species (mol m-3) total initial bulk ionic concentration (mol m-3 ) bulk concentrations of species 0 and R (mol m-3) concentration of species 0 and R at the electrode surface (mol m-3) concentration at distance x and time t (mol m-3 ) total ionic concentration at distance x and time t (mol m-3) concentration of species j at distance x and time t jmol m-3) concentration of species 0 and R at distance x and time t (mol m-3) Laplace transform of co (x, t ) and cR (x, t ) (mol m-3 s-l) steady-state total ionic concentration at distance x (mol m-3) steady-state concentration of species j at distance x (mol m-3) surface concentrations of species 0 and R at time t (mol m-3) diffusion coefficient (m2 s - l )

diffusion coefficient of species j ( m2 s- ) diffusion coefficient of species 0 and R (m2 s- l ) electrode potential (vs. some reference electrode) ( V ) null or equilibrium potential (V) standard potential ( V ) interelectrode potential (V) voltage input and output of a circuit (V)


i ko kf , k ,

L 2 M M M n +

m r i i , Iti

N

N A n

electron Faraday's constant (96 485 C mol-') denotes a functional relationship constant given by eqn. (161) faradaic current (A) jth and (j + 1)th faradaic current values (A) limiting faradaic current (A) faradaic current during the forward and reverse branches of a cyclic voltammogram (A) exchange current (A) transport-free current (Sect. 5.5) (A) faradaic current as a function of time (A) flux (mol m-2 s-') flux of species j (mol m-2 s-' ) flux at distance 3c at time t (mol m-2 s-') flux of species j at distance x at time t (mol m-'s-l) fluxes of species 0 and R at distance x at time t (mol m-2 s-l) steady-state flux of species j (mol m-2 s-l) counter or index, j = 1,2,3, . . . N standard heterogeneous rate constant (m s-' ) heterogeneous rate constants for forward and backward reactions (m s-l) distance between two parallel electrodes (m) Laplace transformation symbol molecular weight (kg mol- ) metal metal ion semi-integral of the faradaic current (A s112) faradaic semi-integral during the forward and reverse branches of a cyclic voltammogram (A s112)

total number of ionic species (Sect. 4.0); upper limit of counter (Sect. 5.4) Avogadro's constant (6.02205 x mol-') number of faradays to reduce one mole of the reducible species oxidized species modified oxidized species at the electrode surface arbitrary function in Laplace space [eqn. (142)] arbitrary function in Laplace space charge ( C ) unit charge (C)

reduced species resistance (ohm)

gas constant (8.31441 J mol-' K-' 1

89

input resistance to an operational amplifier (ohm) j th and ( j - 1)th resistances in a circuit (ohm) radius (m) modified reduced species at the electrode surface denotes "solid", as in M(s) dummy variable in Laplace space (s-l) thermodynamic temperature (K) time interval in a cyclic experiment (s) time (s) lower time limit of semi-integration (s) mobility of an ionic species (m2 s-' V-') mobility of a particular ionic species j (m2 s-' V-l) average velocity (m s-') average velocity at time t (m s-l) local electric field (V m-l) distance measured normal to the electrode surface (m) charge number charge number of a particular ionic species j transfer coefficient surface excess concentration (mol m - 2 ) surface excess concentrations of species 0 and R (mol m-2) activity coefficient constants given by eqns. (75) and (94) time interval (s) energy required to carry an ion from distance x = 0 to x = x (J) viscosity (kg m-l s-l) overpotential ( V ) equivalent ionic conductivity of an ion (m2 ohm-' equiv. - ) order of a derivative (Table 6) electrochemical potential (J mol- ' ) standard electrochemical potential (J mol-') local ionic strength at distance x and time t (mol m-3) local ionic strength at steady state (mol m-3) order of a derivative (Table 6) stoichiometric coefficient of species j support ratio integration variable (s) time of sudden change in imposed conditions after an experiment commences; transition time in a chronopotentiometric experiment (s) local potential (V) reference potential (V)


90

4(x, t ) @(XI

local potential at a distance x and time t ( V ) local potential in the steady state (V)

2. Modes of transport

There are three distinct mechanisms by which electroactive solutes from the solution may reach the electrode or, conversely, by which electrogenerated solutes may be transferred into the solution. These are migration, diffusion, and convection.

Migration is perhaps the easiest to understand. The presence of an electric field causes charged particles to move along the field lines; this is migration. The force experienced by the particle is proportional to its charge and t o the electric field (i.e. the electrical potential gradient). Migration affects ions only; neutral species do not migrate.

Diffusion does not occur in response to any physical force; it occurs because an inhomogeneous solution seeks t o maximize its entropy. Explained another way, the Brownian motion that all solute particles undergo inevitably tends to enrich regions of low local concentration at the expense of neighbouring regions of greater concentration. Diffusion affects charged and uncharged species equally; both ions and molecules diffuse.

In transport by migration or diffusion, the solute particle moves through a stationary solvent. Convection is a totally different process in which the solution as a whole is transported. Solute species reach or leave the vicinity of the electrode by being entrained in a moving solution.

A concise summary of the distinctions between migration, diffusion, and convection is that migration occurs in response to a potential gradient, diffusion in response t o a concentration gradient, and convection in response t o a pressure gradient.

2.1 CONVECTIVE TRANSPORT

We may distinguish two kinds of convection [ 15, 161 : forced convection and natural convection. Forced convection is the result of some motion deliberately introduced by the experimenter; natural convection arises from changes brought about as a result of the electrolysis itself.

Stirring the solution, rotating the electrode, bubbling gases, and pumping solution towards or across the electrode are all methods of inducing forced convection. Such methods yield irreproducible results unless careful attention is paid to the geometry of the system and to ensuring uniformity of the imparted motion. Rotating disks and wall- jet devices are among the most reproducible examples of forced convection . These are the subject of Chap. 5 and will not be further discussed here.

91

The most usual sequence of events leading to natural convection starts when electrolysis generates a region close to the electrode in which a concentration is significantly enhanced or depleted. Usually, a solution that is enriched in a solute has a slightly greater density than the bulk solution and therefore, under the influence of gravity, there is a tendency for enriched regions to fall. Conversely, a region of depleted concentration tends to rise as a result of its slightly diminished density.

The extent to which a solution is able to respond to these gravitational tendencies depends on the geometry of the cell (see Sect. 4). Even under the worst conditions, however, the driving force for natural convection is small and it takes a considerable amount of time for this small force to overcome the inertia of the solution mass. Accordingly, natural convection is unimportant in rapid experiments. In fact, its effect seldom appears before about 10 or 20 s, which is long in relation to most electrochemical experiments.

Henceforth in this chapter, convection will be ignored. In other words, we shall treat only experiments in which forced convection is absent and in which natural convection is inhibited either by judicious geometric design of the cell or by the brevity of the experiment.

2.2 FLUX

The number of moles of a solute being transported across unit area of a surface in unit time is known as the flux at that surface; it has units of mol m-* s-'. The flux, J , may be equated to the average velocity, z), of the individual solute molecules in the direction normal to the surface multiplied by their concentration, c

J ( x , t ) = vc(x , t ) (9) If the surface in question is the electrode and the solute in question is

the electrogenerated species R, then the flux is equal to the rate of generation of R, i.e. the rate of the electrode reaction

Rate of reaction = JR (0, t ) (10) Similarly, the flux of the electroactive species 0 at the electrode surface is given by

(11) where the negative sign arises because of the convention that flux is treated as positive when it occurs in the direction of increasing x , i.e. away from the electrode.

Rate of reaction = - J o (0 , t )

The third component involved in the electrode reaction

0 (soln) + n e - R (soln) (12)

is electrons. Their flux at the electrode surface must also be proportional to the rate of reaction in accordance with Faraday's law and eqn. (2).

References p p . 141-1 43

92

These results together lead to the very important relationship

in which the sign of the current i( t ) reflects our choice of cathodic current as positive.

Expression (13) implies that, for every n electrons that are withdrawn from the cathode, one molecule (or ion) of R is transported away from the electrode. This may be untrue transiently if there is significant adsorption of R. More precisely, expression (13) will be invalid whenever the amounts of adsorption rR and ro of R and 0, respectively, (see Sect. 1.2) are changing significantly with time. Correcting for this effect leads to

Unless we state otherwise, it will henceforth be assumed that the derivatives in expression (14) are negligible, so that expression (13) can be used.

2.3 LAWS OF MIGRATION

Consider a region of solution in which the local potential (I changes along the x axis. Then, - a@/ax denotes the local electric field X . This field acts upon any charge q to produce a force qX acting along the x axis. If the charge is that on an ion of charge number z , so that q = z F / N A , where NA is Avogadro’s constant, then

zFX Electrostatic force = -

NA

If the ion is in a fluid medium, then the electrostatic force seeking to accelerate the ion is opposed by a viscous force trying to slow the ion down. Though it strictly applies only to spheres of macroscopic dimensions, Stoke’s law can provide an approximate expression

Viscous force = 67~.17rv(t) (16) for the viscous drag, 77 being the coefficient of viscosity, r the ion’s radius and v(t) its velocity an instant t.

If MINA is the mass of the ion, then from Newton’s second law of motion

M dv z F X - - = electrostatic force - viscous force = - - 6 n r p ( t ) NA d t NA

time constant

Fig. 8. Velocity versus time curve for an ion in solution.

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On integration, one finds

v(t) = z F X [ 1 - exp [ - 6 7 ~ 2 q r t ) ] 67rNAqr

if the ion was initially stationary. This equation corresponds to an ion experiencing an initial acceleration of zFX/M but settling down to a steady limiting velocity of z F X / ( 6nN,qr) after times long in comparison with the time constant (M/67rNAqr) as shown in Fig. 8.

For common inorganic ions in aqueous solution, one can take the typical values M N 0.1 kg mol-’ , q 5 1.0 x kg rn-l s-l , z = k (1 or 2), r 5 0 . 3 n m and calculate a time constant of the order of 3 x s. Evidently, the limiting velocity is acquired virtually instantaneously, so that the adjective “limiting” may be dropped and v ( t ) contracted to v.

Using the same typical values gives the (velocity/field) ratio as

This quantity is known as the mobility of the ion and is given the symbol u . Some experimental values are given in Table 2 for ions in dilute aqueous solution.

Mobility is accorded a positive sign irrespective of the sign of the ion’s charge, though of course the direction of motion does depend on the sign, cations travelling with the field X (i.e. down the potential gradient a@/ax) and anions in the opposite direction. These signs are taken into account in the equation

z u x z u a $ v = - = - - -

lz I 121 ax

for the velocity of migration. Utilizing identity (9), one may transform this result into the expression


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concentration c f

Fig. 9. Illustration of diffusion “down” a concentration gradient.

z u c ( x , t ) a 4 Iz I a x

- J ( q t ) = -

for the migratory flux.

2.4 LAWS OF DIFFUSION

Diffusion occurs “down” a concentration gradient as shown in Fig. 9. Fick demonstrated that the diffusive flux is proportional to the magnitude &/ax of the concentration gradient of the diffusing solute

a J ( x , t ) =- c ( x , t )

ax

The constant of proportionality is negative (because a positive value of &/ax induces a diffusive flux towards negative x values) and its absolute value is called the diffusion coefficient, D

a J(x, t ) = --D ~ c ( x , t )

ax

TABLE 2

Mobilities of various ions at infinite dilution in aqueous solutions at 25’C [ 171

Ion u/10-8rn2s-’ v-’ H + K + Na+ Ca2 +

OH- c1- NO so:-

36.2 7 . 6 5 .2 6.2

20.5 7.9 7.4 8.3

95

TABLE 3

Values of diffusion coefficients for several species in various media at 25OC [ 181

Diffusant D/IO-" m2 s-l Medium

Cd2+ 6.90 0.1 M KN03 7.15 0.1 M KCI 7.90 1.0 M KC1

Zn2+ 6.38 0.1 M KN03 6.20 1.0 M KN03 6.54 0.1 M NaOH

8.67 0.1 M KC1 I 0 3 10.15 0.1 M KCI

9.89 1.0 M KCl Fe(CN)z- 6.50 0.1 M KC1 Ascorbic acid 10.27 0.1 M NaCl

Cd 16.6

Pb2+ 8.28 0.1 M KN03

0 2 22.6 H20a

Hgb Zn 18.9 Hg Pb 14.1 Hg

a Ref. 19. Ref. 20.

Known as Fick's first law, eqn. (23) is one of the most important equations in electrochemistry.

In aqueous solutions, diffusion coefficients typically have values in the vicinity of m2 s-l , but the exact value for a given diffusant depends, to some extent, on the composition of the solution. Some values are included in Table 3.

The transport of a solute in response to a concentration gradient generally perturbs concentrations in the region. Hence, concentration profiles generally change with time. This change is governed by Fick's second law.

Let us first treat the simplest case in which diffusion is occuring along the x direction of a Cartesian coordinate system, with conditions being uniform in the y and z directions. Such a situation, termed planar diffusion*, is depicted in Fig. 10. Consider the small region of volume A& and the concentration changes that occur within this region in the time interval between t and t + dt. We can write

No. of moles No. of moles No. of moles No. of moles = entering across - leaving across in region

at time t -I- d t at time t plane x plane x + dx in region -

(24)

* Also known by the less appropriate term "linear diffusion".


96

Fig. 10. Planar diffusion.

or C(X, t + dt)Adx - C(X, t)A& = J(x, t)Adt - J(x + dx, t)Adt

(25) Rearrangement leads to the conservation equation

a a -c(x,t) + - J ( x , t ) = 0 a t ax

This is a very general relationship, not restricted to diffusion but applicable to any form of transport.

Fick’s second law follows from combining eqns. (23) and (26). It is usually written

a a 2

a t ax2 -c(x, t) = D - c(x, t)

and states that the local concentration changes with time proportionally to its second derivative with distance. Thus, as shown in Fig. 11, a linear concentration profile is stable; a flux occurs, but does not perturb the concentration.

Equation (27) is an expression of Fick’s second law only for cases in which, as in Fig. 10, the flux lines are parallel, so that equiconcentration surfaces are planar. Figure 12 depicts a situation in which this is not the case. Instead, the flux lines diverge and the equiconcentration surfaces are non-planar. The verbal eqn. (24) holds, but its symbolic replacement is now

C(X, t + dt)Adx -c(x, t)Adx = J(x, t)Adt - J ( x + dx)(A -I- dA)dt

(28) whence the conservation equation becomes

ac a J a 1nA - + - + J - = O at ax ax

a a a 1nA -c(x, t ) + - J(x, t) + ~ J ( x , t) = 0 a t a x ax

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concentration c ,r

Fig. 11. A linear concentration gradient.

Fig. 12. Non-planar diffusion.

and Fick’s second law must be modified to

a a 2 a l n A a a t a x2 a x ax - C(X, t ) = D - c(x, t ) + D ___ - C(X, t )

For many simple geometries, the a lnA/ax term is a simple function of x; it equals 2/x, for example, in the case of divergent spherical diffusion [21]. For other geometries, i3 lnA/dx may be a function of time and of other coordinates [ 21-23].

2.5 COMBINED MIGRATION-DIFFUSION

The mobility, u, of an ion expresses the ease with which it responds to a gradient of electric potential. The diffusion coefficient, D, of an ion expresses the ease with which it responds to a gradient of its concentration. One might well imagine that a relationship exists between u and D. This is indeed the case; the relation is

and is known as the Nernst-Einstein equation. We shall derive it in Sect. 4.2.

The mobility u and diffusion coefficient D of an ion are two of the trio of transport parameters of which the third is the equivalent ionic conductivity A. Conductivity is not a topic here and we will merely quote the

References pp . 141-143

98

tripartite relationship

Among the uses of these relations is their value in providing estimates of the diffusion coefficient of an ion (but watch the units!).

We have ruled out convection, but must still treat situations in which transport occurs by the two remaining modes. In such cases, the total flux of an ion contains a migratory contribution and a diffusive contribution. Hence using eqns. (21) and (23)

a zu a ax Iz I ax J(x, t ) = - D - C(X, t ) - __ C(X, t) - @(x, t ) (33)

Making use of the Nernst-Einstein equation, expression ( 3 3 ) can be rewritten for the combined migration-diffusion flux as

ax a Z F

RT J(X, t ) = -D C(X, t) + ~ C(X, t) - @(x, t )

or as

J(x, t ) = - D a

RT ax - c(x, t) - [RT In c(x, t) + zF@(x, t ) ]

(34)

(35)

In thermodynamic treatments of electrolyte solutions, one defines an electrochemical potential of a species at a uniform concentration c and potential @ by

p = po + RT l n y c + zF(@-@O) (36)

where po is the standard electrochemical potential and 4' is a reference potential. If the activity coefficient y can be treated as a constant, we see that eqn. ( 3 5 ) can be contracted to

D a J(x, t ) = - __ c(x, t ) zp (x, t )

RT (37)

in which the gradient of electrochemical potential appears as the driving force for combined migration-diffusion. This is an elegant formulation of mass transport, but one that we shall not have occasion to use.

3. Preliminary considerations of the experiment

Later in this chapter, we shall be concerned with predicting the out- come of various electrochemical experiments in which transport is a dominant influence. Before such a predictive exercise can be tackled, it is

99

necessary to delineate a number of features of the experiment including (a) the transport geometry, (b) the operative mode(s) of transport, (c) the initial condition, if relevant, (d) the electrical constraint, and (e) the kinetic assumption, if any.

The present section is devoted to discussing each of these features and analyzing the various options.

3.1 TRANSPORT GEOMETRY

Determined by the shape of the electrode, the geometry of the space surrounding the electrode has an important effect on the transport problem.

The terms “finite” and “semi-infinite” are often used to describe transport geometries. A finite geometry means that effects generated at the working electrode reach the other electrode or “double back” to affect the behaviour of the working electrode within the duration of the experiment. Because the effects of diffusion and migration propagate only slowly, the finiteness of the transport geometry is usually only manifested in cells with convection [16] or in rather thin cells of the kind discussed in Sect. 4.

A “semi-infinite” transport geometry implies that, for the duration of the experiment, effects generated at the working electrode do not reach the other electrode or the cell walls, so that the transport coordinate behaves as if it were infinite in extent. The prefix “semi” reflects the fact that the geometry appears infinite in one direction (away from the electrode) but not in the other (towards the electrode). Because of the limited duration of most electrochemical experiments, cells may be tiny and yet be appropriately described as having a semi-infinite transport geometry.

Transport geometries may be classified as “simple” or “complex” according to the number of spatial coordinates needed to describe the transport. Simple geometries involve only one coordinate, that directed normal to the electrode surface, whereas complex transport geometries require two or three coordinates for their description.

Only three simple transport geometries are normally encountered : planar, cylindrical, and spherical. These are shown in Fig. 13. In planar transport*, the flux lines are parallel to each other and normal to the electrode. Cylindrical transport occurs with electrodes that are cylindrical, such as wires, or hemicyclindrical: the flux lines converge in the plane which is normal to the cylinder axis but are parallel in planes which include the cylinder axis. Spherical transport is encountered with spherical or hemispherical electrodes, the flux lines being continuations of the radii

* Also known by the less appropriate term “linear transport”.


100

Fig. 13. Simple transport geometries. (a) Planar; (b) cylindrical; (c) spherical. The arrows indicate flux lines to the electrode surface.

Fig. 14. An “inlaid” electrode.

of the sphere. The space surrounding a hanging mercury drop electrode [24], one of the most important practical electrodes, provides an example of spherical transport geometry.

Of the many “complex” transport geometries, some of the most important are those encountered with “inlaid” electrodes (Fig. 14) [25- 301, in which the electrode is planar and embedded in an insulator whose surface is a continuation of the electrode plane. If such an electrode is large, it may be appropriate to treat it as a case of planar transport, with a correction for the “edge effect” [31, 321. For a small inlaid electrode, however, or for experiments of long duration, such an approximation is no longer valid.

The most popular transport geometry is planar and semi-infinite: more derivations deal with this situation than with any other. There are two reasons for this. First, i t is usually the simplest. Second, more complex geometries (spherical, inlaid, etc.) can often be approximated by this geometry with suitable corrections.

3.2 OPERATIVE TRANSPORT MODE

As discussed in Sect. 2, there are three modes of transport: diffusion, migration, and convection. We first rule out convection, not because it is unimportant, but because convective transport is the subject of Chap. 5.

This leaves migration and diffusion. Rarely is it possible to solve transport problems exactly when two modes participate (some of the situations discussed in Sect. 4 provide exceptions). There is, therefore, a strong motivation to work under conditions in which either diffusion or migration can be eliminated. There is no way of obviating diffusion and, consequently, it is usual to remove migratory effects from a transport experiment .

101

To prevent migration from contributing significantly to the transport of an ion (recall that migration never contributes to the transport of uncharged species), it is sufficient to add to the solution an excess of “supporting electrolyte”. Such an addition provides ions that are electro- chemically and chemically inert but which enhance the ionic strength of the solution to such an extent that the contribution of the electroactive ion is negligible. A fifty-fold excess of supporting electrolyte is usually adequate to reduce migration to a negligible effect, though a greater excess should be used when the electroactive ions carry multiple charges.

It may not be obvious exactly how the addition of excess supporting electrolyte effectiveiy nullifies migration. This will be clearer after reading Sect. 4.6, in which the quantitative effect of additions of supporting electrolyte are assessed.

Addition of supporting electrolyte has a number of beneficial effects other than diminution of the migratory component of transport. We mention four. First, it increases the conductivity of the cell solution, thereby improving the performance of the associated electronic circuitry and decreasing the undesirable “ohmic drop” (see Sects. 4.2, 4.5, and 4.6) of the cell. Second, unless the electroactive species are strongly adsorbed, the double layer (Sect. 1.3) will be populated almost exclusively by supporting ions, hence becoming less of an uncontrolled variable in the experiment. Third, whereas in the absence of supporting ions the activity coefficients of the electroactive ions are strong functions of their concentrations, these activity coefficients are rendered almost constant by a swamping excess of supporting electrolyte. Fourth, the presence of excess supporting electrolyte makes the density of the solution much less dependent on the concentration of the electroactive species, with beneficial consequences in inhibiting natural convection (Sect. 2.1).

3.3 THE INITIAL CONDITION

The pre-existing concentrations in an electrochemical cell prior to the start of the experiment naturally affect the subsequent transport behaviour. Before discussing the various options for such initial conditions, it is useful to provide a classification of electrochemical experiments.

We can classify electrochemical experiments as follows:

Equilibrium experiments: current = 0 Steady-state experiments: current = constant # 0;

potential = constant Transient experiments (single phase or multi-phase) : current and/or potential change

with time Periodic ex-per~hents: cured mdpodemWaie penodfk

functions of time

Referenceirpp. 141-143

102

Equilibrium experiments are carried out under zero-current conditions. Yielding no kinetic information, such experiments need not detain us.

Steady-state conditions are characterized by constancy of the electrical variables: potential and current. Such conditions are usually achieved, if at all, after the experiment has passed through a prolonged transient phase. We may distinguish two kinds of steady state. In the first, cell potential and cell current are constant but local concentrations do change with time. This is not a true steady state; the term “pseudo-steady state” being appropriate. From the viewpoint of the initial condition, a pseudo- steady state is no different from a single phase transient experiment. The second kind of steady state is a “true” steady state: nothing whatsoever changes with time. Such a state represents a unique relationship between current and potential, the states through which the system has previously passed being irrelevant. For such a steady-state experiment (as discussed in Sects. 4.2, 4.4, and 4.5) we need not specify an initial condition.

Most electrochemical experiments are transient: the current and/or potential change with time and there is some clearly identifiable instant (usually denoted t = 0) a t which the experiment commences. The initial condition, that existing immediately prior to t = 0, is usually simple and most often corresponds to zero current and uniform concentrations.

By “multi-phase transient experiments”, we mean those such as cyclic voltammetry (see Chap. 3) or current reversal chronopotentiometry [ 331 in which some sudden charge is made in the imposed conditions at some definite time ( t = T , for example) after the experiment commences. To address such multiphase experiments, one usually treats each phase separately, the solution of the first phase providing details of the concentration profiles at t = T , which then serve as the initial conditions for the second phase. Likewise, the final conditions of the second phase provide the initial conditions for the third phase, if any, and so on.

In periodic experiments, the electrical constraint (the potential perhaps) imposed on the cell is cyclic, being repeated over and over again. Eventually, the cell’s response (the current, for example) settles down

p o t e n t i a l E cu r ren t I A A

0 0

(a) (b)

Fig. 15. A potential step experiment. (a) Potential waveform; (b) current flow versus time.

103

and also becomes cyclic. Alternating current methods (see Chap. 4) constitute the simplest kind of periodic experiment. In a periodic experiment, conditions repreat cyclically so that at time t + T , where T is the “period”, conditions exactly duplicate those at time t. Thus, over an interval of time T , the initial condition and the final condition are identical.

The situation regarding initial conditions may be summarized as follows.

Type of experiment Initial condition

Steady state experiment Irrelevant Single-phase transient or first

phase of a multiphase experiment Usually uniform concentrations

transient experiment Subsequent phases of a multiphase

Periodic experiment Final condition of previous phase Identical with final condition

3.4 ELECTRICAL CONSTRAINT

Though it is possible to conceive of experiments that do not exactly fit either, electrochemical methods generally fall into one of two categories: those in which a potential is imposed on the cell, the current being monitored and those in which a current is applied, the potential being monitored. The first category is the more common. The term “voltammetry” is applied to both categories since each involves a study of how the potential and current are interrelated.

The simplest controlled potential experiment is the potential step [ 341 illustrated in Fig. 15. Such experiments are sometimes termed “chrono- amperometry”, signifying that the current (-ampero-) is measured (-metry) as a function of time (chrono-). Sometimes, two steps, as in a “double-step experiment” [ 341 [Fig. 16(a)], or a sequence of small steps, as in “staircase voltammetry” [35--371 [Fig. 16(b)], are applied. When the potential of the working electrode is changed by a step for only a brief period of time before being returned to its original (or near to its original) value, we speak of a “pulse”. There are many varieties of pulse voltammetry [38-411, some of which are discussed in Chap. 4.

As an alternative to a stepwise variation with time, a continuously changing potential may be imposed. Though other possibilities have been used [42, 431, a linearly changing potential-time waveform, known as a potential ramp [Fig. 17(a)], is the most common. The technique has many names, including “linear sweep voltammetry” [44]. If the direction of the ramp is reversed [Fig. 17(b)], the technique is often termed “cyclic voltammetry” (see Chap. 3), though this name is more appropriately applied after sufficient ramp reversals [Fig. 17(c)] have caused the experiment to become periodic.


104

potential f ,

potential €

1 0 time t 0 > t i m e t

0 0

(a) (b)

Fig. 16. Stepwise potential waveform. (a) Double step; (b) staircase.

E t E

0 ’ -t 0 ;

0 0

t 0 . ;

0

>t

Fig. 1 7 . Potential sweep waveforms. (a) Linear potential ramp; (b) isosceles triangular potential ramp ; (c ) periodic triangular potential ramp.

Through current is the monitored variable in controlled-potential voltammetry, the current-time relationship is not always displayed as such. There are often decided advantages in processing the current and presenting some modified signal as a function of either time or potential. The modifications may be as simple as subtracting the “background” [45] or as complex as Fourier transformation [46-491. Three common processing techniques are integration [ 50-521 , semi-integration [ 53- 601, and semi-differentiation [61-631.

Controlled-current voltammetry generally involves either a sinusoidal waveform (see Chap. 4) or a constant current [Fig. 18(a)]. Constant- current voltammetry, or “chronopotentiometry” [ 64, 651 , generates a potential-time signal, as in Fig. 18( b), characterized by a transition time 7.

The constant current may be reversed in direction at, or before, the transition time, as shown in Fig. 19(a), or repeatedly reversed, thus becoming periodic [Fig. 19(b)]. Less frequently employed is a current waveform which varies as a known function of time [66-691, such as the linear current ramp in Fig 19(c).

105

current 1 potential f

0

(a)

0

(b)

Fig. 18. A chronopotentiometric experiment. (a) Current step; (b) potential vs. time response.

I 1 1

0 0 0

( 0 ) (b) (C)

Fig. 19. Current waveforms. (a) Double step; (b) periodic double step; (c ) linear current ramp.

3.5 KINETIC ASSUMPTION

There are two fundamentally different approaches to utilizing mass transport relationships to determine electrode kinetics. By “determining electrode kinetics”, we mean elucidating the dependence of the rate of the electrode reaction on the variables that affect it. For the reaction

O t n e - R (38)

this generally means determining the form of the function f in the functional relationship

i nAF - - - rate = f(cS,, c;, E ) (39)

where E is the electrode potential and cS, and cg are the surface concentrations of 0 and R at the electrode. Of course, there are other factors (nature of the solvent, temperature, identity of the electrode metal, etc.) on which the reaction rate will depend, but these factors will be constant during any one experiment.

The most direct method of finding f is to amass data giving


106

experimental values of i for various known or calculable values of the variables E , c S , and c& and fitting a mathematical dependence to these data. This direct method has the advantage of not prejudicing the conclusion of the study by preassuming the form of the dependencies of i on E , c& and ck. Despite the superiority of the direct method, it is not the method customarily adopted in electrode kinetic studies.

It is usual to assume that the relationship between the rate of the electrode reaction and the variables E , cS, and c& is given by

I i anF

nAF RT - - - c&k0 exp - __ ( E - E”)

nF

where ko , a, and E” are constants known as the standard rate constant, the transfer coefficient, and the standard (more properly the formal or conditional) potential. Equation (40) is known as the Volmer or B u t l e r Volmer relationship.

The Volmer relationship may acquire simpler limiting forms. If the final term in eqn. (40) is negligible in comparison with the other two, the approximation

i anF

nAF 1 RT x cS, iz” exp - __ (E-E’ )

holds. Reductions conducted under conditions in which this approximation is valid are said to be “irreversible”. Conversely, it often happens that each of the two right-hand terms in eqn. (40) is vastly greater in magnitude than the difference between them, So that

i 0 x cS,koexp

Experiments conducted under such conditions Approximation (42) may be manipulated into

RT ch E = E o - - l n y

nF co

(42)

are termed “reversible”.

(43)

which is the Nernst equation of thermodynamics. Kinetic information is entirely lacking from experiments conducted under reversible or “nernstian” conditions, since the electrode reaction is then at equilibrium.

107

The term "quasireversible" is sometimes employed to denote conditions in which neither the irreversible approximation nor the reversible approximation is valid.

The "null potential", E n , is the potential exhibited by an electrode when equilibrium reigns. The surface concentrations of 0 and R are then identical to their bulk values, c S , = cg and ck = c i , and the current is zero. Solving eqn. (40) for the null potential, we find

This relationship may be inserted back into the general case of eqn. (40) to produce an alternative formulation of the Butler-Volmer relationship.

- -

(45)

( E -En) 1

- - ~ A F K " ( c : ) " ( c ; ) ~ - ~

I n F (1 - a) - ( E -En)

- - 4 ( RT exp

ck

The denominator on the left-hand side of this equation, n A F h " ( c i ) " ( c g ) ' - " , is termed the exchange current, io, while the difference E -En is often called the overpotential q.

4. Finite planar geometry

This section deals with transport between two parallel electrodes. The reaction at one electrode is the converse of that at the other, so that no overall change occurs to the contents of the cell. We consider only the simplest instance of this behaviour in which the anode is a metal, M, dissolved by the reaction

M(s) --n e(M)-M"+(soln) (46)

M"' (soln) + n e(M)- M(s) (47)

the corresponding cathodic reaction being

Thus, there is only one electroactive species: its concentration prior to the electrolysis is uniform and equal to ct . The cell geometry is shown in Fig. 20. Note that, for the reasons discussed in Sect. 2.1, the positioning of the cathode above the anode will act to prevent natural convection. We therefore ignore convective transport but we will, in most of Sect. 4, be concerned with both migration and diffusion.

Because we admit migration, we must be concerned with all ionic species present. We consider a total of N ionic species, and utilize sub- scripts to distinguish them. Thus, we let c!, c';, . . . ch be the hitial


108

area A

Fig. 20. The cell geometry considered in the text.

uniform concentrations of the electroinactive ionic species. Any neutral species present are, of course, unaffected by migration and, since there is no mechanism by which their initially uniform concentrations can be perturbed, no concentration gradients of neutral molecules can be created and hence, no diffusion will occur. Thus, any uncharged species in our solution will be passive bystanders that may be ignored.

Our purpose in considering this particular geometry so early in this chapter is that it is one of the few geometries which may be studied completely. As will be shown in the following sections, the transport problem may be solved exactly, leading to expressions which show how the current, concentrations, and potential vary throughout the cell. Basic principles learned from the investigation of this simple geometry may then be applied to more complicated electrode geometries, which are less easily solved mathematically.

This electrode geometry has also been used as the basis of detection in high performance liquid chromatography [ 70, 711 .

4.1 SOME GENERAL RELATIONSHIPS

In this section, we develop certain general relationships that are always valid and that we shall use later on.

We let cj(x, t ) denote the concentration of ionic species j at a distance x from the cathode at time t . Because each species (electroactive or electroinactive) is conserved, the total number of moles or each species within the cell is a constant and equal to that initially introduced, so that

A f cj(x, t ) d x = ALC? j = 1,2,. . . N (48) 0

gives the total number of moles of species j .

concentration, defined by We shall find it convenient to let ci(x, t ) represent the local total ionic

109

A general relationship that will hold at all times and at all positions in the cell, except within the double layers (see Sect. 1.2), is the electroneutrality condition which reflects the fact that positive and negative charges must occur in equal numbers in any region of macroscopic dimensions. Stated mathematically, this means

N

0 = c Z j C j ( X , t ) j = 1

where zj is the charge number of the species (e.g. zj = - 2 for SO:-). The local “ionic strength” p(x, t ) of the solution is defined by

and this equation is seen to complete a trio of which eqns. (49) and (50) are the first two members.

Just as c j ( x , t ) represents the local concentration of species j , so we use Jj(x, t ) to represent the local flux of ionic species j at time t . From eqn. (34 ) , these two quantities are interrelated by the complicated equation

a Z j F a - C j ( X , t ) + --Cj(X, t ) - @(x , t )

Dj ax RT ax - 4 ( x , t ) -___ -

Recall (Sect. 2.3) that - a$J/ax is the local electric field in the solution: this may vary both in space and time.

Equation (52) contains right-hand terms representing diffusion and migration. The overall effects of these two transport processes may be separated in the following way. Write eqn. (52 ) for j = 1, j = 2, . . . j = N and then sum these equations. The summation generates a migratory term with a factor Zzjcj (x , t ) which eqn. (50 ) shows to be zero. Hence

Likewise, we can cancel out the diffusive terms by multiplying each eqn. ( 5 2 ) by zj before summing. The result is

Because the same current, i ( t ) , flows across both anode and cathode, Faraday’s law tells us that

References pp. 141-143

110

as explained in Sect. 2.2. There are, of course, no fluxes of the electroinactive ions across the electrodes

j = 2 , 3 , . . . N Jj(0, t ) = Jj(L, t ) = 0 (56) though such fluxes may be non-zero elsewhere.

The most general problem would be to prescribe an electrical constraint and solve the above system of equations to predict how the concentration c j ( x , t ) of each ionic species varies, in space and time, from its initial value cj(O < x < L , 0 ) = c; . That problem is inordinately difficult and we must be content with solutions to simpler problems.

4.2 TWO-ION CASE: STEADY STATE

When the electrolyte contains only one ion, necessarily an anion, in addition to the electroactive ion M” +, the electroneutrality condition reduces to the simple

Z l C l ( x , t ) + Z Z C Z ( X , t ) = 0 (57) The results attaching to the steady state may be deduced as special cases of the equations developed in Sect. 4.4, but it is instructive to develop these results ab initio for this so-called “unsupported” case. To begin, we shall not even assume the Nernst-Einstein relationship (Sect. 2.5) between the diffusion coefficient and mobility.

Each ion experiences both a migratory and a diffusive flux and therefore

This equation applies to ion j = 1 or j = 2, both in the steady state and in the transient state that precedes it. In the ultimate steady state, all dependence on time must have disappeared, so the general result reduces to

d zj uj d J j ( x ) = - Dj - c j ( x ) - - c j ( x ) ~ p ( x ) j = 1,2

dx Izj I (59)

One can argue from the conservation equation (26) that, in the steady state, because concentrations do not change with time, the fluxes do not change with distance. Hence Jj ( x ) in eqn. (59) is a constant, independent of x. The same conclusion can be arried at non-mathematically by thinking carefully about the implications of a steady state. Since J j ( x ) is not a function of x , it must equal its value at x = 0 or L , as given in eqns. (55) and (56).

111

Let us first examine the electroinactive ion. From eqn. (59), using the fact that z2 is negative, we have

A verbal interpretation of this equation is that the electroinactive ion experiences a diffusive flux and a migratory flux, which are exactly equal and opposite everywhere in the cell in the steady state. The equation may be rearranged to

and integrated to

D, In c, ( x ) = u, 4 ( x ) + constant If we evaluate the constant of integration by reference to conditions at the cathode, i.e. at x = 0, then

results and can be converted to

This equation describes an equilibrium distribution of the electroinactive ion in an electrostatic field. This is a typical situation to which one may apply the Boltzman distribution law, which states

where NA is Avogadro's constant and E(O + x ) is the energy required to carry an ion from x = 0 to x = x . This energy is simply z2q0 [@(x) - 4(0)] where go is the unit charge, equal to FIN,. When the exponents in eqns. (64) and (65) are equated, the result

"4z2qo - z2F - IzzlF - - - 3 _ _ NA€(O+x)

-

0 2 RT [W) - @ ( O ) l RT RT RT

(66) is obtained. Since all terms in this result are constants, we have derived what must be a general result: it is in fact the Nernst-Einstein relationship discused in Sect. 2.5. By using this relationship, eqn. (59) may be recast as


112

c,(x)

d zjF d Ji - C j ( X ) + - Cj(X) - $(x) = - - dx RT dx Dj

..... .. ....... -

where J , ( x ) has been replaced by Jj because of the constancy of this term. Writing eqn. (67) first for the electroactive ion

d z1F d Jl 2 c1(x) - $(x) = - - = ____ - c1(x) + -

dx RT dx D1 z ~ A F D I

and then for the electroinactive ion

d 22F d JZ

dx RT dx D2 c * ( x ) + - c2(x) -@(x) = - - = 0 -

gives a pair of equations which sum to the simple result

(69)

because condition (57) permits cancellation of the terms involving d$/dx. Using the latter electroneutrality condition once more leads to

d z 2 i - c1(x) = - = a positive constant dx 21 (21 - 22)AFDI

showing that, in the steady state, the ions have adopted a linear concentration profile, as depicted in Fig. 21. One easily shows that the concentrations at the cathode and anode are

and

respectively. Note that, because cl(0) cannot be negative, eqn. (72) tells us that 221 ( 2 1 - z 2 ) A F D 1 c;/lzz IL is the maximum steady-state current the cell will sustain.

113

As our final task in this subsection, let us determine what the potential difference is between the layers of solution adjacent to the anode and the cathode (or, more exactly, between the outer regions of the double layers at the two electrodes, see Sect. 1.2). For this purpose, we turn to eqn. (63), set x = L , and replace D 2 / u 2 by the Nernst-Einstein term RT/ Iz2 IF. Thence

Of course, c 2 ( L ) / c 2 ( 0 ) must equal c l (L) /c , (0) and this ratio can be determined from eqns. (72) and (73). With y1 representing the constant Iz,liL/2zl(zl -z2)AFD1c7, this gives

where we employ the standard definition of an inverse hyperbolic tangent [72] . Note that, in general, the potential difference @(L) - @ ( O ) is not proportional to the current i, i.e. Ohm’s law is not obeyed. Ohmic behaviour does apply, however, when y l , is small for then arctanh y1 may be approximated by Y ~ , leading to

= resistance x R T L @ ( L ) - @ ( O )

i zl(zl - z ~ ) A F ~ D ~ c ~ (76) L

(zl - z 2 ) A F u 1 c ~ - -

Notice that, although relationship (74) superficially resembles the Nernst equation, it differs from this equation in two important respects. First, it relates t o the inactive ion. Second, it gives a potential difference inside the solution, not across the electrodes as shown in Fig. 22. If the electrode reactions are sufficiently rapid so that Nernst’s equation is obeyed (which is not necessarily the case), the actual interelectrode potential difference would be

if double layer effects can be ignored. If Iz2 1 = zl, we see that exactly half the interelectrode potential E , - E , arises from the thermodynamic nernstian effect and the other half from the potential difference generated in the solution by the passage of current. This latter moiety is often


114

Fig. 22. An illustration of the potential difference inside the solution.

termed the “ohmic drop” but this name is unfortunate because, as we have seen, Ohm’s law is not necessarily obeyed.

4.3 TWO-ION CASE: TRANSIENT BEHAVIOUR

Prior to the establishment of the steady states discussed in Sect. 4.2, the cell must pass through transient states which are more difficult to treat. Moreover, to delineate the transient behaviour, we would need to specify the exact electrical constraints. We shall therefore content our- selves here with deriving the general relationship that is obeyed during the transient phase, but we shall not solve this relationship.

The general flux, eqn. (52), may be written separately for the two speciesj = 1 and j = 2

a ZlF a J l ( x , t ) - c1 ( x , t ) + - c 1 ( x , t ) --@(x, t ) a x RT a x D1

= - -

and

a z2F a J 2 k t ) - c 2 ( x , t ) + - c2(x , t ) - @ ( x , t ) a x RT a x D2

= - - (79)

On addition of these equations, the electroneutrality condition, eqn. (57) , causes the terms in &$/ax to cancel, leaving

Differentiation of this sum with respect to x followed by exploitation of the conservation eqn. (26) leads to

Finally, we replace c2(x , t ) by use of eqn. (57), which generates the equation

115

after rearrangement. Equation (82) is to be compared with Fic

a a t

CI(X, t ) = --c1(x, t ) a 2

D1 __ ax2

:'s second law [eqn. (27) J

that would apply to the electroactive M"' ion if it were subject to diffusive transport only. The equations are seen to be identical except that D, in eqn. (83) has been replaced by a composite diffusion coefficient (21 - Z ~ ) D ~ D ~ / ( Z ~ D ~ - z z D 2 ) . This means that the transport of M"+ in the total absence of supporting electrolyte mirrors exactly the transport of M"' when supporting electrolyte is present in excess. Yet, perhaps surprisingly, the intermediate case of modest supporting electrolyte concentration obeys different, and much more complicated, transport laws [73].

We need develop the unsupported finite case no further. Apart from the modified diffusion coefficient, the results developed in Sect. 4.6 apply t o concentration profiles in the two-ion case.

4.4 MULTI-ION CASE: STEADY-STATE CONCENTRATION PROFILES

Transport by combined migration-diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviow with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case.

Since all time dependencies disappear in the steady state, the general equations (53) and (54) reduce to

and

Recall that ci(x) and ~ ( x ) are the local total ionic concentration and the local ionic strength, respectively. Moreover we can argue, as we did in Sect. 4.2, that each J j ( x ) must be independent of x and that all these terms must equal zero, except J , (x) which equals the constant - i/zl AF. Hence


116

Fig. 23. Steady-state concentration profile for the multi-ion situation.

d 1 -- q ( x ) = dx z , A F D ,

and

Because the right side of eqn. (86) is a positive constant, it follows that the total ionic concentration, ci, is a linear function of distance, as shown in Fig. 23. By integration of this equation, one can readily show that

i C i ( X ) = cp +

z ~ A F D I

where cb is the total initial bulk ionic concentration. Though the total ionic concentration is a linear function of distance, it does not necessarily follow that the individual ions have linear concentration profiles. In fact, as we shall see, the anions are linearly distributed, but not the cations.

To proceed much further without complicated mathematics, it is necessary to make a further restriction. We assume that all ions present have the same magnitude of charge

Izi I = z all values of j (89) That is, our analysis henceforth will apply to steady states in solutions containing only monovalent ions, (e.g. T1+, H’, NO;, CH,COO-, etc.) or only divalent ions (e.g. CdZ+, Mg2+, Ca2+, SO$-, etc.), but not mixtures of univalent and divalent ions. The advantage of this so-called “homovalent” restriction is that it provides a simple proportionality

(90)

between the ionic strength and the total ionic concentration, thereby enabling eqn. (60) to be expressed as

117

1 x - (L/2) -I- (zAFD,c;/i) (91)

The second equality in this expression is, of course, a consequence of eqn. (88)-

We are now in a position to consider individual ions, rather than the ions collectively. The steady state flux equations for ions of charge magnitude z are

- - I z F d RT dx '(x) = zAF Dlci(x) --

-ccj(.) d k -cCj(x) z F - d $(XI = - - Ji dx Di dx R T

(92) i

j = 1 zAFD1 =(a j = 2 , 3 , . . . . N

where the upper sign applies to cations and the lower to anions. Substi- tution from eqn. (91) now leads to

1 j = 1

j = 2 , 3 , . . .N

(93)

d 1 --cj(x) f dx x - ( L / z ) + ( Z A F D ~ C ~ / ~ )

in which x and ci(x) are now the only variables. Equation (93) is, in fact, a simple ordinary differential equation that is soluble by standard methods [74] to give

constant - j = 1

L x - ( L / 2 ) + ( W y )

Cj(X) = constant x - - + - i : 2 3

constant

j = electroinactive anion

(94) j = electro-

inactive cation

where we use y as an abbreviation for iL/2zAFDI c:. The integration constant in each of the three alternatives is evaluable by recourse to the general relationship (48) and the standard integral

(95)


118

C j ( X ) = ‘

The final results

.p ( 2 2 L

c; - -

‘ electroactive cation

-Y + 1) electroinactive anions

Cj” electroinactive cations

[ ( 2 x / L ) - 1 + (l/7)] arctanh y I (96) give the concentration profiles for the three kinds of hornovalent ions. Figure 24 shows typical examples of these concentration profiles.

Of special interest are the values of the M”’ ion concentration at the cathode and anode. These are

and

where p is an important quantity known as the “support ratio” and is defined by

concentrations

0 distance L

Fig. 24. Steady-state concentration profiles for the three kinds of homovalent ions in solution. The graph was drawn for cy = 1 and cb/2 = 1.75.

119

concentration of electroinactive ions excluding counter ions P = concentration of electroactive ions plus counter ions

where counter ions are those anions that are essential to neutralize the elec troac tive cations.

4.5 MULTI-ION CASE: STEADY-STATE POTENTIAL PROFILE

The distribution of potential in the solution when a homovalent steady state exists is of considerable interest. Obtained by integration of eqn. (91), it is

- @(x) = constant + In z F

RT

where, as before, y abbreviates iL /2zAFD1 cb . Figure 25 shows a typical example of a potential profile.

The total potential drop in the solution, from the layer adjacent to the anode to that adjacent to the cathode, is given by

RT l + y 2 R T

Z F 1-7 z F - arctanh y @ ( L ) - @ ( O ) = - In ~ - -

and is seen to depend totally on the magnitude of the parameter y. Table 4 shows some values of the potential drop. Notice that for y up to about 0.5, the potential drop is approximately proportional to y, so that the behaviour is close to ohmic with

R T L small y - - 2 R T y

= resistance = ~

@ ( L ) - @ ( O ) i zFi Z ~ A F ~ D ~ C :

( 1 0 2 )

potent 1 o 1

0 d i stance L

potent 1 o 1

0 d i stance L

Fig. 25. Potential profile for a hornovalent steady state, drawn for y = 1/3.

References pp. 141-1 43

120

TABLE 4

Values of the potential drop at 25OC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99 1.00

0 5.2

10.4 15.9 21.8 28.2 35.6 44.6 56.4 75.6 94.1

136.0 W

4.6 ROLE OF SUPPORTING ELECTROLYTE

We are now in a position to examine, in quantitative detail, the effect of the supporting electrolyte, at least for steady-state homovalent transport in finite planar geometry. The amount of supporting electrolyte present, relative to electroactive electrolyte, is characterized by the support ratio that was defined as

in eqn. (99). The value of p can range from zero, when there is no supporting electrolyte (the two-ion case of Sect. 4.2 and 4.3), to positive values approaching infinity, when there is a large excess of supporting electrolyte.

The value of the support ratio places an upper limit on values that can be acquired by y, the parameter that played such an important role in Sects. 4.4 and 4.5. That this is so is evident from eqn. (97) when it is appreciated that the cathodic concentration c, (0) of M”’ can never be less than zero. Hence

l + P 7 - < -

(1 - y)* arctanh y P

From this inequality, it is possible to calculate the maximum y value corresponding to any given support ratio, as has been done in constructing Table 5. For large values of the support ratio, a series expansion in the form

121

TABLE 5

Maximum y value calculated using eqn. (104) corresponding to a given support ratio

P Maximum

M

100 50 20 10

5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0

0 0.00497 0.00987 0.0242 0.0469 0.0883 0.188 0.304 0.443 0.623 0.733 0.816 0.889 0.925 1

1 + ( 1 / 3 p ) - (19/72p2) + . . . P (105) -

2 ( 1 + P ) Ymax -

holds. Now, the existence of a maximum to y implies a maximum value to

the steady-state current i. Such a maximum is called a limiting current and its value, from the definition of y, is

2 z A F D l c ~ y m , 42AFD1(1 + p ) c i ymax (106) - - lM =

L L

The second equality in eqn. (106) follows from eqn. (103), the definition of the support ratio. A graph showing the dependence of the limiting current on the support ratio is shown as Fig. 26. The limiting current when p = 0, the completely “unsupported” case, is given by

4zAFDlc’; L

ifii,(p = 0 ) =

an expression which is identical t o that derived in Sect. 4.2. For values of p large enough that we may use expression (105) with only two numeratorial terms (i.e. p 2 10, approximately), then

2zAFDlcy [ 3:] ifi, (large p ) = 1 + -

L

Thus the value of the limiting current given by eqn. (107) for the totally


122

I I

1 i m i t ing cu r ren t i l l m

Fig. 26. Limiting current as a function of support ratio. Current has been normalized by division by 22 A F D1 cy /L .

unsupported case is exactly twice that given by eqn. (108) when the supporting electrolyte is in infinite excess. This factor of two arises because the limiting current in the absence of supporting electrolyte is composed to two components, a diffusive component and a migratory component, that, in the homovalent case, are exactly equal in magnitude. Addition of supporting electrolyte progressively diminishes the migratory component so that, in the limit of infinite support, only the diffusive moiety remains.

Equation (108) can provide an answer to the important question of how much supporting electrolyte it is necessary to add to effectively inhibit migration. If “effectively inhibit migration” is interpreted to mean “reduce the migratory component to 1% of the diffusive component”, we see that the support ratio needs to be 33. Thus, a 33-fold excess of supporting electrolyte renders migration negligible, at least in the homovalent steady state. There is reason to believe that, in a heterovalent situation, the supporting electrolyte must contribute 97% of the total ionic strength to nullify migration to 1%.

A much-debated question is that of how the addition of supporting electrolyte serves to inhibit migration. Sometimes, one reads that it is because the current is largely carried by the supporting ions (i.e. the transport number of the electroactive ion is rendered negligibly small). But this explanation is clearly untenable: in the steady state discussed in Sects. 4.4 and 4.5, the electroactive ion carries all the current: the other ions are immobile. One must seek an explanation in terms of the potentials existing in the solution. By placing a limit on the value of y, a large support ratio restricts the potential drop that can exist in solution between the outer layers of the electrodes. Combining eqns. (102) and (105), we have, approximately

123

Thus, with a support ratio of 26 or more, the potential drop can never exceed 1 mV. I t is the absence of a potential gradient to drive migration that makes diffusion dominant.

The situation considered in Sects. 4.4-4.6 is one of the few cases in which the effect of supporting electrolyte can be examined quantitatively. Nevertheless, it is reasonable to assume that the same principles hcld in situations, transient as well as steady state, that cannot be treated exactly. Certainly the attitude of electrochemists to migration is “add supporting electrolyte and forget it”. This will be our attitude in most of the rest of the chapter.

4.7 TRANSIENT BEHAVIOUR TO A CURRENT STEP

We shall develop relationships in this seciton on the assumption that sufficient excess of supporting electrolyte is present so that migration can be ignored. However, making allowance for a changed effective diffusion coefficient, exactly the same relationships would hold in the total absence of supporting electrolyte. The situation with intermediate levels of supporting electrolyte would not, however, be the same.

The electrical constraint that we here consider is the imposition of a constant current on the cell commencing at time t = 0

Prior to the imposition of the current step, the solution is uniform so that, for ion M”+

c1(x, t ) = c; 0 < x < L , t < 0 (111) Subsequent to t = 0, Fick’s second law is obeyed

(112)

For t > 0, condition (I 10) is equivalent to imposing a constant concentration gradient on each end of the cell

(113)

because of relationship (55) and Fick’s first law. The set of equations (111)-( 113) fully delineates the transport

problem. The equations are deceptively simple and their solution is by no means trivial. Here, we shall omit details of the solution technique, referring the reader to the original literature [75 ] . The solution, which expresses the concentration of the Mu+ ion as a function of distance and time, can be written in two forms


124

concentration t

L I I 0 L /2 L

Fig. 27. Concentration profiles predicted by eqns. (114) and (115). Curves were drawn assuming L = 0.001 m, A = 0.001 m2, c’: = 1 mol m-3, D = 1 X lo-’ m2 s-’, i = 1 X A, and n = 2 . Time increments are 10 s and 50 s.

j L - L + x c1(x, t ) = c’; + -

nAF 2i J“ D1 i = 1 , 2 (--y [ierfc ( w j L - x

- ierfc ( ~ ) j 2 4 i E

where ierfc denotes the first integral of the error function complement [ 7 6 ] , and

i

n A F D , c1(x , t ) = c’; - ~

(115) 4L 1

Astonishingly, these two disparate expressions describe exactly the same function. Some concentration profiles predicted by these equations are shown in Fig. 27.

Equations (114) and (115) predict that the cathode and anode concentrations are symmetrically disposed about the original bulk concentration, c ; , (which, incidentally, remains unperturbed at the centre of the cell, x = L / 2 )

c’r - C l ( 0 , t ) = c1 ( L , t ) - c’;

125

The two series in this expression converge so quickly that, to within a precision of 0.2%

L2 t < __

20D1

C I ( 0 , t ) c1 ( L , t ) 8 I-- = - 4 c; llim

(117) where i, is the limiting current 2nAFDl c: /L .

As would be expected, the cell eventually reaches a steady state if the imposed current is less than the limiting current, the surface concentrations being

and

Figure 28 shows an example of the way these surface concentrations change in the approach to the steady state. The time that it takes for the surface concentrations to reach within 1% of their final steady-state values can be calculated from eqn. (117) as 0.45 L2 /Dl .

foncen t ra t 1 on

I t I I Q 0 0. 1 0. 2 0. 3

Fig. 28. Change in the electrode surface concentrations as the steady state is approached. Time is normalized by division by L’JD.

References pp . 141-1 43

126

If the applied current exceeds the limiting current, the above equations are obeyed, but only up to a specific time, termed the transition time r. This is the time at which the concentration of Mn + at the cathode reaches zero. The transition time is calculable from eqn. (117) as

r = {

ifim < i < 1.85 ifi,

After the transition time, reaction (47) can no longer sustain the current i and the potential of the cathode will shift negatively until some other electrode reaction (such as solvent reduction) becomes feasible.

4.8 OTHER EXPERIMENTS WITH FINITE PLANAR GEOMETRY

Constant-current experiments, such as that described in the preceding subsection, are not as popular as experiments in which the potential is the controlled variable. In this section, we shall address controlled- potential experiments in cells with finite planar geometry, even though such experiments do not lend themselves readily to kinetic studies. We shall assume, as in previous sections, that the anode reaction is the converse of the cathode reaction.

Mn++ne-M (121)

so that the total amount of electroactive material remains constant and equal to its initial value. I t will also be assumed that sufficient supporting electrolyte is present to nullify the migration of Mn+ and to make the "ohmic drop'' (Sect. 4.2) negligible.

In a controlled-potential experiment, it is not possible, without making an assumption about the electron-transfer process, to predict the faradaic current or to predict the concentration of M"' at the surfaces of the cathode or anode. One can, however, write the simple equation

Cl(0, t ) + c,(L, t ) = 2 q (122)

interrelating the two surface concentrations, and the relationships

9 { c 1 ( 0 , t ) } = - " - tanh (i E ) s n A F m

in Laplace space [77, 781 linking the surface concentrations to the current. We shall omit the derivation of these results.

127

The above equations are valid for any experiment in a cell with finite planar geometry. For example, they apply to the experiment described in Sect. 4.7; in fact, eqns. (116) can be derived from eqns. (123) and (124) by setting i(t) equal to the constant i and performing a Laplace inversion. The Laplace inversion is difficult in this derivation and the interested reader is referred to ref. 79 for guidance.

For reasons already mentioned in Sect. 3.1, cells with finite planar geometry are usually thin cells (i.e. L is small, usually a fraction of a millimetre) and there are only two electrodes. A controlled-potential experiment thus usually involves fixing the potential between the two electrodes, though this does not necessarily mean fixing the potential ucross either electrode. That is, the way in which the applied potential divides itself between the anode and the cathode will, in general, change with time, even if the total applied potential remains constant. For this reason, the simplification that normally attends experiments carried out a t constant applied potential is not achieved with finite planar cells.

I t is only when the electron transfer occurs reversibly (Sect. 3.5) that it is possible to predict the time dependence of the current generated by a finite planar cell in response to an applied potential step of magnitude E. Under reversible conditions, a constant applied potential engenders the constant surface concentrations [ 8 0 ]

and

- 2 c’2

1 + exp (nFE/RT) c1 (L, t ) =

where E represents the (negative) potential of the cathode with respect to the anode.

The Laplace inversion of eqn. (123) or eqn. (124) when the surface concentrations are constant is accomplished rather easily and leads to

m

i ( t ) = nAF(c? - c ; ) d 3 [ 1 + 2 c exp (- ”)] (127) nt j = 1 401 t

or, equivalently

2nAFD1 L

i ( t ) =

where c; is the constant concentration at the cathode surface given by eqn. (125). The first of these equations is more useful at short times, the second at long times. For example, as t + 00, eqn. (128) reduces to


128

2nAFDI

L i(t) = (c? -

As expected, this result is identical to the t = 00 version of eqn. (116): the same steady state is achieved whether the experiment employs a constant potential or a constant current.

5. Semi-infinite geometry

In semi-infinite planar geometry, the electrode occupies the 3t = 0 plane and transport occurs perpendicularly to that plane from a limitless unimpeded medium as shown in Fig. 29. To prevent radial diffusion to the edge of the electrode, it is necessary to have “walls” of some kind to constrain the transport direction to be normal to the electrode. Because of this requirement, electrodes with precise semi-infinite planar geometry are difficult to fabricate and are, in fact, rather rare in practice. Never- theless, because theoretical derivations are simplest for this geometry and because many practical geometries closely approximate the semi-infinite planar one as a limiting case, the geometry of this section is of paramount importance.

We shall be concerned almost exclusively with the reaction

O f n e e R (130)

occurring a t a planar electrode of constant area A . The initial condition will be a uniform concentration & of the electroreducible species 0 throughout the solution. If R is present initially, its concentration is also uniform and of value &. For cases in which R is a metal dissolved in mercury (see Sect. l.l), the mercury phase is also regarded as semi- infinite. Our treatment will be restricted to instances in which both 0 and R are transported solely by diffusion, with diffusion coefficients Do and DR, respectively.

In Sect. 5.6, we briefly describe modifictions needed when the semi- infinite geometry is spherical rather than planar.

5.1 A SIMPLIFYING RESULT

Under the conditions that have been specified, the electroactive species 0 and R each satisfy Fick’s second law

Fig. 29. Semi-infinite planar geometry.

129

(131)

with the attendant initial and asymptotic boundary conditions

co(x, 0) = C O ( ~ , t) = cg

C R ( x , 0 ) = CR(co, t ) = c$

(133)

(134)

Moreover, relationship (13) ensures that the fluxes of 0 and R at the electrode surface are equal and opposite, so that

As will be proved in Sect. 5.3, one can demonstrate that a direct consequence of the above system of equations is that the individual concentrations of species 0 and R are necessarily linearly related through the equation

= constant (136)

This simple relationship means that, if the concentration profile of 0 is known, then the corresponding R concentration profile is readily calculated. N o comparable relationship exists for most other geometries. For x = 0, we have the very simple result

- c&] = a [c; -&I (137)

Because Do and DR are often very similar in magnitude, relationship (136) means that the total concentration of 0 plus R remains approximately constant at all times and in all locations in the electrolyte solution.

5.2 SURFACE CONCENTRATIONS

Another simplying result that applies (in exact form) only to semi- infinite planar geometries relates the surface concentrations of 0 and R directly to the faradaic current i ( t ) . The relationship, which is proved in Sect. 5.3, is

References pp. 141-143

130

TABLE 6

Definitions and properties of the semi-operators

Definitions

Properties

d” d” - c ajfj(t) = c uj - fj(t) d t ” j = l j = l d t”

d” dV - f (b t ) = b ” - d t ” [d(bt)]” f (b t )

m d”

d t” j = 1

d” d’l d”+’l f ( t ) = - _ _ _

d t ” dt’l d t v + ’ l f ( t )

Linearity

Scale change

Leibnitz’s rule

Composition rule for p < 1 and f (0 ) = 0

and is a direct consequence of eqns. (131)-(135) coupled with Faraday’s law. The important consequence of result (138) is that, for electrochemical reactions at planar semi-infinite electrodes uncomplicated by homogeneous chemical reactions, there is never a need to consider the variation of concentration with distance. The surface concentrations, which alone are important in electrode kinetics, are directly linked to the current: the x variable does not appear in eqn. (138) and can be ignored.

Equivalent to relationship (138) is its inverse 1 d-1/2 -- i(t) = 6 [ C h -c&(t)] = flR [ c d ( t ) - c i l nAF d t - ’”

(139) which shows how surface concentrations may be calculated directly from faradaic current. Note that eqn. (139) incorporates result (137).

The d’/2/dt’/2 and d- 1/2/dt-1/2 operators are respectively the semi- differentiation and semi-integration operators [ 811 . These are analogues of the familiar differentiation and integration operators of the calculus. Since they are unfamiliar to many chemists, Table 6 has been included to illustrate some of their definitions and properties. The semi-

131

TABLE 7

Examples of the semi-operations

f d1I2 f

dt'" __

- 112 f

#j t - 112

0 0

C C, any constant __

@

t

I 0

0

4 t3'2

3G __

1 - exp ( b t ) erf (@?) fi

Jz + fi exp ( b t ) erf (\/i;z)

differentiation and semi-integration operators become more tangible when applied to well-known functions and some examples are tabulated in Table 7. In Sect. 5.4, we shall explain how these operations may be carried out in practice.

5.3 DERIVATIONS

Here, the techniques of Laplace transformation are used to derive the

Taking the Laplace transform of Fick's second law, eqn. (131), yields important results reported in Sects. 5.1 and 5.2.

d2 - Do - CO ( x , S ) = sCO ( x , S ) - CO ( x , 0 ) = sFO ( x , S ) - C; (140) dx2

where Co (x, s) denotes the Laplace transform

of the concentration co (x , t ) of the oxidized species 0. The initial condition (133), namely co(x, 0) = c$ was used in the final step of eqn. (140).

Equation (140) is a second-order ordinary differential equation and standard methods [ 741 show that its solution must be of the form


132

(142) where P o ( s ) and Q o ( s ) represent arbitrary functions of the “dummy” variable s. The second arbitrary function, Q o ( s ) must, however, be zero because otherwise Co (x, s) would approach infinity as x + 00 whereas, in fact, transformation of eqn. (133) shows that Co (00,s) = cg/s.

Accordingly

4 Eo (x, s) = - -Po (s) exp

S

and, by exactly similar reasoning

Differentiation of these equations leads to

and

(143)

Equations (145) and (146) apply at all points in the electrolyte solution. Let us, however, specialize them to x = 0, multiply each by the corresponding diffusion coefficient, and add, leading to

From eqn. (135), however, the left-hand side of eqn. (147) must be zero. I t follows that

Returning, with this new result, to eqns. (143) and (144), we can write

133

and

If we now adjust the distance scale for species R relative to species 0 so that

(151) then addition to eqn. (149) gives

(152) which, on Laplace inversion, provides a proof of eqn. (136).

Next, lei? us combine eqns. (143) and (145) into

by elimination of the unknown Po (s) function. Because semi- differentiation with respect to time is equivalent, in Laplace space, to multiplication by 6 inversion of eqn. (153) leads to

Similarly

Finally, we specialize eqns. (154) and (155) to x = 0 and utilize eqns. (13) and (23) to produce eqn. (138). Semi-integration of eqn. (138) leads directly to eqn. (139).

5.4 SEMI-INTEGRATION IN PRACTICE

To exploit the equation


134

from Sect. 5.2, one needs some practical method of semi-integrating current with respect to time: we discuss such methods in this section. The symbol m is used to denote the semi-integral of the current, which is also known by the less satisfactory name of "convoluted current" [ 57-59]. Notice that, as in integration, a lower limit must be specified if the semi- integral is to be unambiguously defined. The operator symbol

implies that the lower limit (the time at which the semi-integration commences) is t = 0. For a lower limit other than zero, the standard notation is

d-112

[d( t - to)] - 1'2

where to is the lower limit. Two approaches have been used to semi-integrate electrochemical

currents: analog [43, 82-84] and digital [55, 60, 851. Each has been used satisfactorily in experimental voltammetry. We discuss analog methods first.

In order to appreciate the principles of analog semi-integration, let us first review the classical operational amplifier circuits shown in Fig. 30. The output of circuit (a) i s a value of the current input

Eout = -Ri

whereas the output of circuit current

voltage proportional to the instantaneous

(159)

(b) is proportional to the integral of the

In eqn. (160), we write the unusual operator notation d-lldt- ' for indefinite integration to emphasize the analogy with semi-integration.

When the feedback loop of the operational amplifier contains a resistor,

R

(a) (b) Fig. 30. Classical operational amplifier circuits. (a) A current follower; (b) a current integrator .

135

Fig. 31. A geometric ladder network.

as in Fig. 30(a), the output is proportional to i. When the feedback element is a capacitor, as in Fig. 30(b), the output is proportional to the integral of i. One might surmise that a circuit element suitably intermediate between a resistor and a capacitor would engender an output proportional to the semi-integral of i. I t turns out that this surmise is correct and that the “suitably intermediate circuit element” is well approximated by the so-called “geometric ladder” illustrated in Fig. 3 1. The components in this network are selected so that the ratio Ri/Cj of the jth resistance to the jth capacitance is a constant, R/C, and the ratio

of the adjacent elements is another constant g, typically equal t o m When such an element is placed in the feedback loop of ari operational amplifier, as in Fig. 32, the output is a voltage closely proportional to the semi-integral of the input current

Other methods exist for analog semi-integration [86-881, but Fig. 32 adequately illustrates the principles. Note that this circuit transforms a current input into a voltage output. With the symbol shown as Fig. 33 serving as an abbreviation for the ladder network, the circuit shown in Fig. 34 serves as a voltage input-voltage output semi-integrator with an output given as - p E O “ ,

Fig. 32. A current semi-integrator.


136

Fig. 33. Symbol used to represent a ladder network.

Fig. 34. Semi-integrating circuit.

Fig. 35. Semi-differentiating circuit.

The related circuit, shown in Fig. 35, performs semi-differentiation d1/2

Eout = - Ci, - d t 1/2 Ein (164)

and such a device has been incorporated into commercial electrochemical instrumentation [ 891 .

Turning now to digital methods of semi-integration, we shall describe a technique by which the semi-integral rn rnay be determined from a set of equally spaced current values, i,, i,, i,, . . . , i j , . . . , i N . If A is the time interval between adjacent points and the first datum, io, corresponds to t = 0, the ij corresponds to the time instant j A and iN corresponds to t = N A . Digital semi-integration enables a value of m(t ) , equal to m(NA) or simply mN to be calculated from all the values io , i,, . . . , iN .

The so-called Reimann-Liouville definition [ 811 of a semi-integral is

This integral may be split into N components

If A is small enough, one can, with little error, approximate the current between two consecutive measured values by assuming a linear variation

i(r) = ( j + l - a ) i j + (a - j ) i j+l j A < T < A + j A (167)

137

c

Fig. 36. Linear interpolation used in digital semi-integration.

between the values. Fig. 36 shows the significance of this interpolation procedure. When eqn. (167) is inserted into eqn. (166) and the indicated integrations are carried out, one finally arrives at

I N - 1

+ 1 ij { ( N - I - ~ I ~ ’ ~ - z ( ~ - j ) ~ / ~ + ( N + I - ~ ) ~ / ~ ) j = l

This formula, termed the RL-algorithm, based as it is on a “connect-the- dots” approximation, is the semi-integration equivalent of the trapezoidal formula of integration,

The RL-algorithm is a good general-purpose formula for semi- integration. There are circumstances, however, in which it is less than ideal. One such circumstance occurs following a potential step, where the resulting current is a continuously declining function of time. In such experiments, the initial current, i, , is usually experimentally inaccessible and may even be theoretically infinite. Special algorithms [ 90, 911 have been devised to handle such situations.

5.5 EVALUATION OF ELECTRODE KINETICS BY THE DIRECT METHOD

As explained in Sect. 3.5, an investigation of the kinetics of the electrode reaction

O + n e =+ R (169) by the direct method boils down to deciphering the form of the function linking the rate of the electron transfer to the relevent concentration and potential variables

i n A F

= rate = f(cS,, c&, E ) (170)

At a planar electrode, however, in the absence of complications arising


138

from other chemical reactions, there are the simple relationships between the surface concentrations c& and c& and the semi-integral m [eqn. (156)]. These relationships are linear and involve only constants, so that eqn. (170) may be replaced by

1 - = rate = f(m, E ) nAF

Thus, the kinetics can be elucidated by a study that involves only the electrical variables (potential and current) together with the semi-integral of the current.

In a potentiostatic experiment, E is held constant so that, according to eqn. (170), the rate of the electrode reaction depends only on the surface concentrations of the species 0 and R. One of the simplest circumstances that can be imagined would have the reaction occurring bidirectionally with the forward rate proportional to the concentration of 0 and the backward rate proportional to the concentration of R , so that

where hf and kb are the two rate constants. These two constants will take definite values at an invariant potential, but are expected to be dependent on potential in the sense that hf will become larger but h, smaller as the potential is made more negative and electrons thereby become more available to effect reduction. If eqn. (172) is to apply at equilibrium, then the two rate constants must be in the ratio

in order that the Nernst equation be obeyed when rate = 0. Hence, combining eqns. (156), (172), and (173), we arrive at the experession

(174) for the forward rate constant. This result shows how the rate constant may be calculated from purely electrical variables ( E , i, m ) and constants.

A similar approach is that adopted by Bond et al. [92]. These authors used cyclic voltammetry (Sect. 3.4) as a means of obtaining two values each of the current (7' and r) and its semi-integral ( 2 and &) at each potential E. They demonstrated that these four values could be combined into a quantity

139

which is the faradaic current that would have flowed across the working electrode at potential E if there were no transport impediment whatsoever. In other words, instead of eqn. (172), one has

:*

in which the concentration terms have their bulk values. Since cyclic voltammetry provides values of i* over a range of potentials, E , during a single experiment, eqn. (176) provides a simple route to the determination of the potential dependence of the rate constants kf and kb*

5.6 SPHERICAL ELECTRODES

Through, for reasons of mathematical simplicity, planar electrodes are assumed in most voltammetric derivations, they are seldom used in practice because of the difficulties of their fabrication. The favorite, and most convenient, electrode of experimental voltammetrists is the static mercury drop. This spherical electrode cannot always be adequately approximated by a planar model. In particular, eqn. (139) does not hold for a spherical electrode and hence semi-integration cannot be used to generate accurate values of surface concentrations. Indeed, even eqn. (137) is invalid, the concentrations of 0 and R not being linearly related a t the surface of a spherical electrode.

A t a spherical electrode, one must consider a spherical diffusion field as discussed in Sect. 2.4. Fick’s second law is then written

and

at all times and for x > 0. Here, r is the radius of the spherical electrode of area A and x is the distance coordinate outward from and normal to the electrode surface. Assuming that both species, 0 and R, are solution- soluble, the initial and asymptotic boundary conditions

c o ( x , 0 ) = co (m, t ) = cg

CR ( x , 0 ) = CR (w, t ) = ck

(179)

(180)

and


140

must also be obeyed. Following mathematics which closely parallels that presented in Sect. 5.3, it is possible to derive the following relationships for the case of a spherical electrode that replace expression (139). When R dissolves in the solution, the equations are

and

where fs is the function

1 fs (y) = ___ - exp (Y 1 erfc (fi) &T

However, when R amalgamates, Fick’s second law and its attendant conditions take the forms

which, after much mathematics, leads to the expression

which replaces eqn. (182), though eqn. (181) still applies. Here, fa is a more complicated function not to be discussed here (see ref. 93 for details).

The integrals in eqns. (181), (182), and (187) are technically known as convolution integrals; the procedure which uses them has been called “spherical convolution” [ 931. Spherical convolution may be used to determine the concentrations of 0 and R at the surface of a spherical electrode in a fashion analogous to the use of semi-integration to determine concentrations ‘at the surface of a planar electrode.

Recall that at a planar electrode, one operation, namely the semi- integration of the current, provides the information from which both concentrations may be calculated. At a spherical electrode, two distinct operations are necessary; one to generate cS, and a second to generate

141

c i . The only exception arises when R dissolves in solution and has a diffusion coefficient equal to that of 0. In this exceptional case, eqn. (137) is valid, even for a spherical electrode.

Acknowledgements

The financial assistance of the Natural Sciences and Engineering Research Council of Canada is acknowledged with gratitude.

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