CHAPTER 19 ELECTROCHEMICAL KINETICS

19.1 Introduction................................................................................................................................. 1

19.2 Electron Transfer at Metal Electrodes...................................................................................... 3

19.2.1 Electron Transfer and Reaction Rate.......................................................................... 319.2.2 Activation Energy for Electron Transfer..................................................................... 519.2.3 The Butler-Volmer Equation...................................................................................... 819.2.4 The Tafel Equations.................................................................................................... 1319.2.5 Mass Transfer Effects................................................................................................. 1519.2.6 Electrical Double Layer Effects.................................................................................. 19

19.3 Redox Electrodes......................................................................................................................... 20

19.3.1 Multi-step Reactions at Metal Electrodes .................................................................. 2019.3.2 Adsorbed Intermediates and Electron Transfer........................................................... 2519.3.3 The Hydrogen Electrode............................................................................................. 2919.3.4 The Oxygen Electrode................................................................................................ 3319.3.5 The Hydrogen Peroxide Electrode.............................................................................. 4119.3.6 The Chlorine Electrode............................................................................................... 41

19.4 Mixed Potentials.......................................................................................................................... 41

19.4.1 Multiple Redox Couples at an Electrode.................................................................... 4119.4.2 Surface-Controlled Kinetics........................................................................................ 4319.4.3 Mixed-Control Kinetics.............................................................................................. 4719.4.4 Transport Controlled Kinetics..................................................................................... 50

19.5 Electron Transfer at Semiconductor Electrodes...................................................................... 57

19.5.1 Energy Levels in Electron and Hole Transfer............................................................. 5719.5.2 Current-Potential Relations......................................................................................... 5919.5.3 Transport of Charge Carriers in Semiconductors....................................................... 6319.5.4 Redox Reactions......................................................................................................... 68

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19.1 Introduction

On the basis of the nature of the reactants and products, three main types of electrode

processes may be identified: (a) ionic redox reactions, (b) gaseous redox reactions, and (c) phase

change reactions.

In an ionic redox electrode process (Figure 19.1a), both the reactant and product are

water-soluble ionic species, although they have different charges. In gaseous redox reactions

(Figure 19.1b), a reactant or a product is a gas. Examples are the oxygen, hydrogen, and the

chlorine electrodes. In a typical gaseous redox process, a hydrated ion or a water molecule

reacts, through adsorbed intermediates, on the electrode surface; dissolved gas molecules are

produced initially, and these subsequently combine to form gas bubbles.

A2+e- +

A+

(a)

A2

e-2 + 2A+

(g)

(b)

M+

e-

M+ H O2

(H O)2 n+M

(c)

M+

Mo

e- +

M Deposit(d)

MX (s)M+

e-

MX-+

MX Deposit(e)

Figure 19.1 Types of electrode processes: (a) ionic redox reactions (b) gaseous redox reactions, (c) electrodissolution, (d) electrodeposition, (e) surface film reactions.

Phase change electrode processes may involve phase formation or dissolution, as

illustrated in Figures 19.1c-e. In the case of electrodissolution (Figure 19.1c), a surface metal

ion detaches and enters the aqueous phase as a hydrated ion; the associated electron is donated to

the electrode. In electrodeposition (Figure 19.1d), a hydrated metal ion receives electrons from

the electrode and the resulting electroneutral metal becomes incorporated into the crystal

structure of the metallic deposit. The metal ion released during a dissolution process may

combine with an aqueous species to give an insoluble product, i.e., a surface film (Figure 19.1e).

In a related process, a previously formed surface film may be made to undergo dissolution.

We shall limit the discussion in this Chapter to electron-transfer reactions in which the

electrode is inert, i.e., cases where the electrode serves only as a source or sink for electrons,

without undergoing any chemical transformation itself. Also, we will not consider here those

cases where there is surface deposition. This means that the focus in this Chapter is primarily on

ionic redox electrodes and gaseous electrode processes. Dissolution processes are discussed in

Chapters 20 and 21 while deposition is treated in Chapters 22 and 23.

19.2 Electron Transfer at Metal Electrodes

19.2.1 Electron Transfer and Reaction Rate

The potential of an electrode is a measure of the energy of the constituent electrons. As

the electrode potential moves in the negative direction, the electrons rise to increasingly high

energy levels. At sufficiently negative potentials, the energy levels occupied by the electrons

become high enough to permit electron transfer from the electrode to aqueous phase species.

Suppose that the working electrode has a potential E relative to a reference electrode. Suppose

further, that at this applied potential, the following reaction occurs at the metal electrode

surface:

Az+ + ne- B (19.1)

That is, electrons from the solid are received by an aqueous species Az+ at the solid/aqueous

interface, and the subsequent reaction yields a product B. This electron transfer constitutes

current flow.

Since the current flow is associated with the Az+/B reaction, it is of interest to establish a

quantitative relationship between reaction rate and current. When a current I flows for a time t

and results in the consumption of nA moles of a species A, Faraday's law gives

nA = It/nF (19.2)

where F is the Faraday constant, i.e., 96487 coulombs/g equiv., and n moles of electrons are

involved in the reaction of 1 g mol of A; n has units of g equiv./g mol.

Let us recall the definition of reaction rate on a unit surface basis (see Equation 15.7),

rA = (1/S)dnA/dt (19.3)

For constant current, Equation 19.2 can be differentiated with respect to time to give,

rA = (1/S)dnA/dt = I/nFS (19.4)

= i/nF (19.5)

kf

kr

where iA is the current density, defined as

i = I/S (19.6)

For the reaction occurring on a metal electrode surface (Equation 19.1), let kf and kr

respectively be the rate constants for the forward and reverse reactions. Then the net reaction

rate is given by

rA = (1/S)dnA/dt= -kfCA + krCB (19.7)

Using Equation 19.6 in Equation 19.5 gives

i = -nFkf CA + nFkrCB (19.8)

Inspection of Equations 19.1 and 19.8 reveals that the forward reaction (which consumes

electrons) contributes a negative current, whereas the reverse reaction (which releases electrons)

contributes a positive current. A convention shall be used such that a positive current is said to

flow when positive charge flows from electrode to solution, i.e., electron generation (anodic

reaction) occurs and electrons flow from solution to electrode. Thus, the forward reaction of

Equation 19.1 is associated with a negative current density since it involves the transfer of Az+

(a positively charged species) from the solution to the electrode, and electrons from the electrode

to the solution, i.e., electrons are consumed (cathodic reaction). On the other hand, the reverse

reaction which transfers a positively charged species to the aqueous phase, and electrons from

the solution to the electrode, gives rise to a positive current.

19.2.2 Activation Energy for Electron Transfer

The activation energy of a chemical reaction can be taken as a constant at a given

temperature. In contrast, the activation energy of an electrochemical reaction is greatly

influenced by the electrode potential. Recalling the electrode reaction described by Equation

19.1, it would be expected that, since Az+ is charged, its reaction would be affected by the

potential difference between the metal electrode (M) and the aqueous solution (Aq). As the

value of (M - Aq = ∆) becomes less positive, the attraction of Az+ to the electrode surface

would be enhanced and therefore the rate of the forward reaction would rise (i.e., k f would

increase). Furthermore, it would be harder to reject Az+ from the electrode surface and therefore

the reverse reaction would slow down (i.e., kr would decrease). This dependence of the rate

constants on the potential difference can be expressed quantitatively in terms of an Arrhenius-

type equation, with an activation energy which is proportional to (M - Aq):

kf = kf,o exp[-(1 -)nF(M - Aq)/RT] (19.9a)

kr = kr,o exp [nF(M - Aq)/RT] (19.9b)

where F is the Faraday constant, is termed the transfer coefficient and gives the fraction of the

potential difference that influences the forward reaction, i.e., (1 - )(M - Aq), as well as the

fraction that affects the reverse reaction, i.e., (M - Aq).

We can gain some insight into the origin of the parameter by considering the free

energy changes associated with the electron transfer process. When a potential is applied to the

electrode, the energy of an electron in the electrode is altered. As illustrated in Figure 19.2, with

a positive potential (E>0), the energy is lowered compared with the E=0 condition. In contrast,

when a negative potential (E<0) is applied, the energy of the electron is raised.

Energy

Potential

(a) E = 0 (b) E > 0 (c) E < 0

Figure 19.2 Relationship between electrode potential and electron energy.

Referring to Figure 19.3, we can say that when the electrode potential is zero (E = 0 V),

the reactants (Az+ + ne-) and the product (B) are associated with free energies which change with

distance as shown by the solid curves. It must be noted that the free energy of Az+ increases as it

approaches the electrode surface since it becomes necessary to (fully or partially) discard the

waters of hydration of this species. Correponding to the cathodic (forward) and anodic (reverse)

reactions of Equation 19.1 are the activation energies G#oc and G#

oa respectively.

When a positive potential (E) is imposed on the electrode, the energy of the electron is

lowered (Figure 19.2b) and consequently, the curve representing the (Az+ + ne-) configuration

moves downwards by the amount nFE, as shown by the dashed curve in Figure 19.3. The

resulting cathodic and anodic activation energies are G#c and G#

a respectively. It can be seen

that application of the positive potential E has the consequence of lowering the anodic activation

energy by a certain fraction (nFE) of the overall energy change. It follows from Figure 19.3

that

G#a = G#

oa - nFE (19.10a)

Examination of Figure 19.3 further shows that:

G#c + nFE = G#

oc + nFE (19.10b)

Thus

G#c = G#

oc + (1-nFE (19.10c)

Standard Free Energy

Reaction Coordinate

+Az+ ne -

E = 0

E = E

B

nFE G #c

G #oc

G #a

G #oa

nFE

nFE

Figure 19.3 Relationship between electrode potential and the activation energy for electron transfer.

We can express the rate constants kf and kr in terms of the following Arrhenius-type

equations:

kf k f ,o exp Gc# RT (19.11a)

kr k r,o exp G a# RT (19.11b)

It follows from Equations 19.10 a, c and 19.11 a, b, that

kf k f ,o exp Goc# 1 nFE RT

k f,o exp Goc# RT exp 1 nFE RT

k f,o exp 1 nFE RT (19.12a)

kr k r,o exp G oa# nFE RT

k r,o exp Goa# RT expnFE RT

k r,o exp nFE RT (19.12b)

where

kf ,o k f ,o exp Goc# RT (19.13a)

kr ,o k r,o exp Goa# RT (19.13b)

Based on Equations 19.12a and 19.12b, Equation 19.7 can be rewritten as:

rA = -kf,oCAexp[-(1-)nFE/RT] + kr,oCBexp[nFE/RT] (19.14)

Also, in view of Equations 19.8, 19.12a, and 19.12b, the current density can be expressed as:

iA = -nFkf,oCAexp[-(1-)nFE/RT] + nFkr,oCBexp[nFE/RT] (19.15)

19.2.3 The Butler-Volmer Equation

At equilibrium, E = Eeq. Also, at equilibrium, rA = 0, i=0, and therefore (recalling

Equations 19.7 and 19.8),

kfCAe k rCBe (19.16)

where CAe and CBe are the corresponding equilibrium concentrations of A and B. For the special

situation where the electrode potential and solution conditions are such that CAe = CBe, Equation

19.16 gives:

kf k r ko (19.17)

where ko is termed the standard rate constant. When CAe = CBe, the corresponding equilibrium

potential is termed the formal potential, E. It follows from Equations 19.12 a,b and 19.17 that

kf ko exp 1 nF E E / RT (19.18a)

k r ko exp nF E E / RT (19.18b)

EXAMPLE 19.1 The Nernst Equation

Starting with the expression derived above for the current density associated with the overall electrode reaction (Equation 19.8), show that

Eeq E RT nF ln CA CB (1)

Solution

From Equation 19.8 we know that

i nFkf CA nFk rCB (19.8)

Also, at equilibrium, i = 0 and therefore kfCAe = krCBe; also, E = Eeq. It follows then from Equations 19.18a,b, and 19.8 that

CAe ko exp 1 nF Eeq E RT CBeko exp nF Eeq E

RT (2)

Thus,

CAe CBe exp 1 nF Eeq E RT exp nF Eeq E

RT (3)

Rearranging, we get the desired expression:

Eeq = E' + (RT/nF)ln(CAe/CBe) (4)

Equation 4 represents the Nernst equation for the Az+/B couple.

Combining Equation 19.8 with Equations 19.18a, b gives:

i nFkoCA exp 1 nF E E RT nFkoCB exp nF E E RT (19.19)

Equation 19.19 can be rewritten as:

i = i- + i+ (19.20)

where i- is the cathodic current, and i+ the anodic current:

i- = -nFkoCAexp[-(1-)nF(E-E')/RT] (19.21a)

and

i+ = nFkoCB exp[nF(E-E')/RT] (19.21b)

At equilibrium, i = 0. Thus, Equation 19.20 gives:

i i io nFkoCBe exp nF Eeq E RT nFkoCAe exp 1 nF Eeq E RT (19.22)

That is, at equilibrium, the anodic and cathodic currents have the same magnitude, i.e., io, termed

the exchange current density. It follows from Equation 19.22 that:

nFk oCAe io exp 1 nF Eeq E RT (19.23a)

nFkoCBe io exp nF Eeq E RT (19.23b)

From 19.21a and 19.23a,

i CA CAe io exp 1 nF E E Eeq E RT

CA CAe io exp 1 nF E Eeq RT (19.24a)

From 19.21b and 19.23b,

i CB CBe io exp nF E E Eeq E RT

CB CBe io exp nF E Eeq RT (19.24b)

Using Equations 19.24a and 19.24b in Equation 19.20,

i io CB CBe exp nF E Eeq RT CA / CAe exp 1 nF E Eeq RT = io{(CB/CBe)exp[nF/RT] - (CA/CAe)exp[-(1-)nF/RT]} (19.25a)

where is termed the activation overpotential and is given by:

n = E - Eeq (19.25b)

Equation 19.25a is called the Butler-Volmer equation. Figures 19.4a and b illustrate

respectively, the variation of (i/io) and log (|i|/io) with the overpotential .

The above analysis indicates that if the rate of electron transfer is slow, then in order to

obtain significant reaction rates, it is necessary to provide an applied potential that is sufficiently

greater than the equilibrium potential.

EXAMPLE 19.2 Relationship between the exchange current density and the standard rate constant

Show that when CAe = CBe = Co,

io = nFkoCo

Solution

Recall the Nernst equation (Ex19.1):

Eeq E RT nF ln CAe CBe (1)

That is,

exp nF Eeq E RT CAe CBe (2)

Multiply Equation 2 by exp [-(1-)]:

exp 1 exp nF Eeq E RT C Ae CBe exp 1

= exp[-(1 - )] (3)

In Equation 3 the condition CAe = CBe has been used.

We know from Equation 19.22 that

io = nFkoCAe exp[-(1 - )nF(Eeq - E')/RT]

Therefore, it follows from Equation 3 that Equation 19.22 can be rewritten as

io = nFkoCAe = nFkoCo

where the condition CAe = Co has been used.

Recall: from 19.8 and 19.20,

i- = - nFkf CA (19.26)

Compare 19.24a and 19.26:

nFkfCA CA CAe io exp 1 nF RT

kf io nFCAe exp 1 nF RT (19.27)

Similarly, from 19.8 and 19.20,

i+ = nFkr CB (19.28)

Compare 19.24b and 19.28:

nFk rCB CB CBe io exp nF E Eeq RT

k r io nFCBe exp nF E Eeq RT

k r io nFCBe exp nF RT (19.29)

Summary

kf k f ,o exp Gc# RT (19.11a)

k f,o exp 1 nFE RT (19.12a)

ko exp 1 nF E E RT (19.18a)

io nFCAe exp 1 nF RT (19.27)

kr k r,o exp G a# RT (19.11b)

(19.12b)

(19.18b)

io nFCBe exp nF RT (19.29)

19.2.4 The Tafel Equations

If || >> (RT/F), two simplified relations called the Tafel equations arise:

(a) When is positive, the second term in Equation 19.25a can be neglected, with the

result that:

i = io exp[nF/RT] (19.30a)

or

ln i = ln io + nF/RT (19.30b)

That is,

ln i = ln (nFCBkr,o exp[nFEeq/RT]) + nF/RT (19.30c)

(b) When is negative, the first term in Equation 19.25a can be neglected to give:

i = -io exp[-(1 - )nF/RT] (19.31a)

or

ln |i| = ln io - (1 - )nF/RT (19.31b)

That is,

ln |i| = ln {nFCAkf,oexp[-(1 - )nFEeq/RT} - (1 - )nF/RT (19.31c)

The linear relationship between log (|i|/io) and implied in Equations 19.30b and 19.31b

can be seen in Figure 19.4b for large values of ||.

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EXAMPLE 19.3 The Fe3+/Fe2+ electrode

Gerischer, Z. Elektrochem., 54, 366 (1950)Lewartowicz, J. Chim. Phys. 49, 564 (1952)Vetter and Manecke, Z. Physik. Chem., 195, 270 (1950)Lewartowicz, J. Chim. Phys. 49, 573 (1952)Petrocelli and Paolucci, J. Electrochem. Soc., 98, 291 (1950)

EXAMPLE 19.4 The Ce4+/Ce3+ electrode

Vetter , Z. Physik. Chem., 196, 260 (1951)Petrocelli and Paolucci, J. Electrochem. Soc., 98, 291 (1950) or 1951?Lewartowicz, J. Chim. Phys. 49, 564, 573 (1952)____________________________________________________________________________________________

19.2.5 Mass Transfer Effects

It was pointed out in Chapter 17 that the motion of an ion in an aqueous electrolyte

solution is controlled by: (a) diffusion caused by a concentration gradient, (b) migration due to

the presence of an electric field, and (c) a convective or hydrodynamic transport due to bulk fluid

motion. In the absence of migration and convection, the rate of transport is determined by the

concentration gradient.

Figure 19.5 Mass transfer in electrode processes

Recall the AZ+/B reaction at the electrode surface:

(19.1)

Referring to Figure 19.5 and considering Equation 19.1:

Flux of A to the surface:

(19.32a)

Flux of B from the surface:

(19.32b)

Az+Az+(s)

B(s) B

(rA)

NA

NB

CAS

CA

CB

Electrode Solution

CBS

kf

kr

Rate of consumption of A by the reaction at the electrode:

rA 1 S dnA dt k fCAS k rCBS i nF (19.32c)

Under steady-state conditions, we must have

N A rA NB i nF (19.33)

From Equation 19.32a,

CAS = CA – (NA/kdA) (19.34)

From Equation 19.32b,

CBS = CB + (NB/kdB) (19.35)

Substituting Equations 19.34 and 19.35 into Equation 19.32c,

rA k f NA kdA CA k r CB NB kdB (19.36)

Recalling Equation 19.33, NA = NB = rA, and thus, Equation 19.36 becomes:

rA kf kdA rA k fCA k rCB k r kdB rA

rA 1 k f kdA k r kdB k fCA krCB

rA kfCA k rCB

1 k f kdA k r kdB (19.37)

It can be seen from Equation 19.32a that the rate of mass transport of A is greatest when

CAS «CA, i.e., (CA - CAs) CA. Under these conditions, any A that reaches the electrode

surface is instantaneously consumed by the reduction reaction. The resulting current density is

termed the cathodic limiting current density, iLc, and recalling Equation 19.33:

iLC nFkdACA (19.38)

Similarly, in the case of the anodic reaction, the rate of mass transport of B (Equation 19.32b) is

greatest when CBs «CB, i.e., (CB - CBS) CB. In this case, any B reaching the electrode surface

is immediately oxidized. The resulting current density is termed the anodic limiting current

density, iLa:

iLa nFkdBCB (19.39)

Writing rA in terms of current density (Equation 19.33), and using Equations 19.38 and

19.39 to substitute for CA and CB respectively, Equation 19.37 becomes:

i nF k f iLC nFkdA k r iLa nFkdB 1 k f kdA k r kdB (19.40a)

i k f kdA iLC k r kdB iLa 1 k f kdA k r kdB (19.40b)

CASE 1: Only A is present in solution. What happens when only A is present in

solution? Under these conditions, CB = 0 iLa = 0. It should be noted that both kf and kr are

potential-dependent. Recalling Equations 19.12a and 19.12b, it can be seen that as the potential

(E) becomes more negative, kf increases while kr decreases. In the absence of B, the potential

must be decreased sufficiently to drive the cathodic reaction. Under these circumstances, k f » kr .

Therefore, using the constraints iLa = 0, and kf » kr, we get from Equation 19.40b:

i kf kdA iLC 1 k f kdA k fiLC k dA kf (19.41a)

Rearranging,

1

i

k dA

k f iLC

1

iLC(19.41b)

Recalling Equation 19.38, (i.e., iLC = - nFkdACA), Equation 19.41b becomes

1 i 1 nFkf CA 1 nFkdACA (19.41c)

Depending on the relative magnitudes of kf and kdA, the electrode reaction will be kinetic

or transport controlled. When kdA » kf, Equation 19.41c becomes:

1 i 1 nFkf CA (19.41d)

or

i nFkf CA (19.41e)

That is, in this case, the reaction is under kinetic control. On the other hand, when kdA «kf,

Equation 19.41c simplifies to:

1 i 1 nFkdACA (19.41f)

or

i nFkdACA (19.41g)

In this case the reaction is transport controlled.

CASE 2: Only B is present in solution. Here CA = 0 iLC = 0. Also kr » kf (for reverse

of reasons given above for Case 1). Therefore, Equation 19.40b simplifies to:

i k r kdB iLa 1 k r kdB k riLa k dB k r (19.42a)

or

1 i k dB k riLa 1 iLa (19.42b)

Recalling Equation 19.39 (i.e., iLa = nFkdBCB), Equation 19.42b may be rearranged as:

1 i 1 nFk rCB 1 nFkdBCB (19.42c)

When kdB » kr, the reaction is under kinetic control:

1 i 1 nFk rCB (19.42d)

i.e.,

i nFkrCB (19.42e)

On the other hand, when kdB « kr, the reaction is under transport control and

1 i 1 nFk dBCB

(19.42f)

i nFkdBCB (19.42g)

19.2.6 Electrical Double Layer Effects

19.3 Redox Electrodes

19.3.1 Multi-step Reactions at Metal Electrodes

As discussed in Chapter 16, an overall chemical reaction will typically consist of two or

more steps. Electrochemical reactions are no exception. Thus, for example, on certain metal

surfaces, the hydrogen evolution reaction involves a two-step process of the form:

H3O+ +e- (M) H (M) + H2O (19.50a)

H (M) + H (M) H2 (19.50b)

The dissolution of a divalent metal ion, e.g., Cu, may involve the following steps:

Cu Cu+ + e- (19.51a)

Cu+ Cu2+ + e- (19.51b)

The Mn4+/Mn3+ redox electrode has the following overall reaction:

Mn3+ = Mn4+ + e- (19.52a)

The reaction steps are:

2Mn3+ Mn4+ + Mn2+ (19.52b)

Mn2+ Mn3+ + e- (19.52c)

In the case of iodine reduction, the overall reaction is:

I3- + 2e- = 3I- (19.53a)

The relevant reaction steps are:

I3- I2 + I-

(19.53b)

I2 2I (19.53c)

I + e- I-

(19.53d)

Consider a general electrochemical reaction in which 2 electrons are transferred from an

electrode to a species A, transforming it to C:

A + 2e- = C (19.60)

As discussed in Chapter 15, we can derive a rate law for this reaction with the aid of the steady-

state assumption or the rapid equilibrium assumption.

Suppose this reaction proceeds via the following single-electron transfer steps:

Steady-state Assumption

A + e- (k1, k-1) B (Step 1) (19.61)

B + e- (k2, k-2) C (Step 2) (19.62)

Following the procedures developed in Chapter 15, we can write:

rA = -k1CA + k-1CB (19.67)

rB = k1CA - k-1CB - k2CB + k-2CC (19.68)

rC = k2CB - k-2CC (19.69)

As found previously in Chapter 15, applying the steady-state approximation to reactive

intermediate B (i.e., rB = 0) gives:

CB = (k1CA + k-2CC)/(k-1 + k2) (19.70)

Inserting Equation 19.70 into Equations 19.67 and 19.69,

rA = (-k1k2CA + k-1k-2CC)/(k-1 + k2) (19.71)

rC = (k1k2CA - k-1k-2CC)/(k-1 + k2) (19.72)

Bearing in mind the convention adopted here, i.e., a reaction that releases an electron

generates positive current, the net current associated with the overall reaction can be expressed

as:

i/F = -k1CA + k-1CB - k2CB + k-2CC (19.73)

Using Equation 19.70 to substitute for CB in Equation 19.73 gives:

i/F = 2(-k1k2CA + k-1k-2CC)/(k-1 + k2) (19.74)

It can be seen by comparing Equations 19.71 and 19.74 that i/F= 2rA. This is consistent with the

fact that the overall reaction involves the transfer of two electrons and that when the steady-state

approximation is valid, the rates of the successive steps are equal. Recalling Equations 19.12a

and 19.12b, the rate constants can be expressed as:

k1 = k1,o exp[-(1-1)FE/RT] (19.75)

k-1 = k-1,o exp[1FE/RT] (19.76)

k2 = k2,o exp[-(1-2)FE/RT] (19.77)

k-2 = k-2,o exp[2FE/RT] (19.78)

It is instructive to consider approximate forms of Equations 19.71, 19.72, and 19.74. For

example, when k-2 << k1, and k2 << k-1, we get:

CB = k1k2CA/k-1 (19.79)

rA = -k1k2CA/k-1 (19.80)

rC = k1k2CA/k-1 (19.81)

i/F = -2k1k2CA/k-1 (19.82)

Recalling Equations 19.75, 19.76, and 19.77, Equation 19.80 can be rewritten as:

(19.83a)

(19.83b)

where

(19.84)

(19.85)

Accordingly, the net current density becomes:

(19.86)

The parameter (=1+2) is termed an apparent transfer coefficient.

Rapid Equilibrium Assumption. Let us now consider the case where the first electron

transfer involves a rapid equilibrium:

(fast) (19.89)

(slow) (19.90)

The rapid equilibrium means that the forward and reverse rates of Equation 19.89 are equal in k1

magnitude:

(19.91)

Also,

(19.92)

It follows from Equation 19.91 that

(19.93)

Inserting Equation 19.93 into Equation 19.92,

(19.94)

Referring to the stoichiometry of the overall reaction (Equation 19.60), production of 1 mole of

CA is associated with the generation of two moles of electrons. Thus:

(19.95)

It can be seen that Equations 19.93, 19.94, and 19.95 are identical to Equations 19.79,

19.81, and 19.82 respectively. Thus, applying the conditions k-2<<k2 and k2<<k-1 to Equations

19.61 and 19.62 is equivalent to replacing Equation 19.62 by the irreversible reaction of

Equation 19.90.

19.3.2 Adsorbed Intermediates and Electron Transfer

In the discussion above, the Butler-Volmer equation was derived on the assumption that

the entire metal electrode surface was accessible to the reactants in the aqueous solution. In fact,

there are many situations where reaction intermediates adsorb on the electron surface, thereby

decreasing the effective surface available for reaction. Referring to Equation 19.1, suppose now

that the product B is retained as an adsorbed intermediate on the electrode surface:

Az+ + ne- B (ads) (19.110)

Then recalling Equation 19.14, the corrresponding rate equation can be written as:

rA = -CA(1 - B)kf,oexp[-(1 - )nFE/RT] + Bkr,oexp[nFE/RT] (19.111a)

where B is the fraction of the electrode surface that is occupied by the adsorbed intermediate B.

Similarly, recalling Equation 19.15, the current density now becomes:

i = -nFCA(1 - B)kf,oexp[-(1 - )nFE/RT] + nFBkr,oexp[nFE/RT] (19.111b)

At equilibrium, the exchange current density is given by:

io = i+ = nFBekr,oexp[nFEeq/RT]  (19.112a)

= -i- = nFCA(1 - Be)kf,oexp[-(1 -)nFEeq/RT] (19.112b)

where Be is the equilibrium surface coverage of B. Thus, we can rewrite Equation 19.111b as:

i = -nFCA[(1 - B)/(1 - Be)](1 - Be)kf,oexp[-(1 - )nFE/RT]

+ nF[B/Be]Bekr,oexp[nFE/RT] (19.113a)

= -io [B/Be]exp[-(1 - )nF(E - Eeq)/RT]

+ io[(1 - B)/(1 - Be)] exp[nF(E - Eeq)/RT] (19.113b)

= io {[B/Be]exp[nF/RT] - [(1 - B)/(1 - Be)]exp [-(1 - )nF/RT]} (19.113c)

The Langmuir Adsorption Isotherm. Also, it follows from Equations 19.112a and

19.112b that

/(1 - ) = CAKexp[-nFEeq/RT]

(19.114)

where K = kf,o/kr,o. Equation 19.114 represents a Langmuir adsorption isotherm for the

intermediate B. The standard free energy of adsorption is given by

Gads = -RTlnK (19.115)

The Temkin Adsorption Isotherm. The Langmuir adsorption isotherm is based on the

assumption that the free energy of adsorption is independent of the surface coverage. However,

there is experimental evidence indicating that this is not generally the case. The coverage-

dependent adsorption has a number of origins, including interactions between adsorbed species,

heterogeneity of surface sites, and surface-dipole effects. According to the Temkin treatment,

the surface is a composite of Langmuir adsorption sites, each with its own characteristic standard

freee energy of adsorption. It is assumed that the standard free energy of adsorption varies

linearly with coverage:

Gads, = Gads,0 + r (19.116)

where r is a proportionality constant and Gads,0 represents the standard free energy of

adsorption at zero coverage. It should be noted that since the free energy of adsorption is a

negative quantity, Equation 19.116 shows that with r positive, the magnitude of the free energy

of adsorption decreases with coverage. It follows from Equation 19.116 that

∂Gads,/∂ = r (19.117)

Similarly, in the case of the free energy of desorption, we can write:

Gdes, = Gdes,0 + r' (19.118)

where Gdes, = -Gads,, Gdes,0 = -Gads,0, and r' = -r.

Associated with the coverage-dependence of the free energy of adsorption is a coverage-

dependence of the activation energy of adsorption (∆G*ads,). In general, it is assumed that the

two are related linearly, i.e.,

∂G*ads,/∂ = ∂Gads,/∂ = r (19.119)

Similarly, for the activation energy of desorption,

∂G*des,/∂ = ∂Gdes,/∂ = 'r' = -(1 - )r (19.120)

where the proportionality constant ' is related to as:

' = (1 - ) (19.121)

It can be seen from Equation 19.121 that has the characteristics of a symmetry factor.

It follows from Equations 19.119 and 19.120 that:

G*ads, = G*ads,0 + r (19.122)

G*des, = G*des,0 + 'r' = G*des,0 - (1 - )r (19.123)

Returning now to Equation 19.110, we can write the rate equation for the Temkin condition as:

i = -nFCA(1 - B)k'f,oexp[-(1-)nFE/RT]exp[-G*ads,/RT]

+ nFBk'r,oexp[nFE/RT]exp[-G*des,/RT] (19.124)

Using Equations 19.122 and 19.123 respectively to substitute for G*ads,and G*des,in

Equation 19.124 gives:

i = -nFCA(1 - B)kf,oexp[-(1-)nFE/RT]exp[-r/RT]

+ nFBkr,oexp[nFE/RT]exp[(1 - )r/RT] (19.125)

For intermediate coverages, the contributions of the pre-exponential terms in are small

compared with the corresponding exponential terms. Thus, Equation 19.125 may be simplified

to:

i = -nFCAkf,oexp[-(1-)nFE/RT]exp[-r/RT]

+ nFkr,oexp[nFE/RT]exp[(1 - )r/RT] (19.126)

At equilibrium, i = 0, and Equation 19.126 reduces to:

= -nFEeq/RTr + (1/r)ln KCA (19.127)

For purposes of comparison, the Langmuir adsorption isotherm (Equation 19.114) may be re-

expressed in logarithmic form as:

ln ln (1 - ) = -nFEeq/RT + ln KCA (19.128)

Examination of Equations 19.127 and 19.128 indicates that as CA is increased, increases more

rapidly with the Langmuir isotherm compared with the Temkin isotherm; this effect increases

with increase in the magnitude of the proportionality constant, r. This situation can be

understood physically by recognizing that as the coverage increases, further adsorption becoms

less favorable, because of the corresponding decrease in |∆G|.

19.3.3 The Hydrogen Electrode

In Chapter 10, it was pointed out that water decomposes under reducing conditions to

give molecular hydrogen, and under oxidizing conditions to give molecular oxygen. These

reactions frequently serve as the half-reactions for many technologically important

electrochemical processes. They may also represent competing reactions which therefore

decrease the efficiency of the reactions of interest. Thus, in oxygen-free solutions metals with

standard reduction potentials below that of the standard hydrogen electrode will dissolve, with

evolution of hydrogen:

M + 2H+ M2+ + H2 (19.129)

This reaction can be decoupled into its anodic and cathodic half-reactions as:

M M2+ + 2e- (19.130)

2H+ + 2e- H2 (19.131)

The cathodic evolution of hydrogen is also of importance in metal deposition systems where it

represents a competing reaction to the metal reduction reaction.

Pathways for the Hydrogen Evolution Reaction. Two main mechanisms are accepted for

the hydrogen evolution reaction. These are the chemical-desorption mechanism:

H3O+ + e- H (ad) + H2O (19.132)

H (ad) + H (ad) H2 (19.133)

and the electrochemical-desorption mechanism:

H3O+ + e- H (ad) + H2O (19.132)

H (ad) + H3O+ + e- H2 + H2O (19.134)

It can be seen from Equations 19.132-19.134 that in both cases, the first step is the

reduction of a hydrated proton to give an adsorbed hydrogen atom. However, the two

mechanisms differ in what happens next. In the chemical-desorption mechanism, the adsorbed

hydrogen atoms diffuse over the electrode surface. When two such atoms encounter one another,

they react chemically to give a hydrogen molecule, which then desorbs (Equation 19.133). In the

case of the electrochemical-desorption pathway, additional protons are reduced onto the first

layer of adsorbed hydrogen atoms. Atoms from the first and second layers then combine to form

hydrogen molecules (Equation 19.134). For either mechanism, the rate-determining step may be

the the first step (case a) or the second step (case b).

The chemical-desorption mechanism. Let us first consider case (a), i.e., the situation

where the first step (i.e., the discharge step) is rate-determining:

H3O+ + e- H (ad) + H2O k1 (19.135)

H (ad) + H (ad) H2 k2, k-2 (19.136)

The current density associated with the rate-determining step is given by:

i1 = Fk1,0[H3O+](1 - H)exp[-(1 - )FE/RT] (19.137)

For relatively low surface concentration of adsorbed atomic hydrogen, H 0 and Equation

19.137 simplifies to:

i1 = Fk1,0[H3O+]exp[-(1 - )FE/RT] (19.138)

Now, the entire two-step reaction involves the transfer of two electrons. Thus, the net current is

given by:

i = 2i1 = 2Fk1[H3O+]exp[-(1 - )F/RT] (19.139)

Alternatively, for case (b), where the second step (i.e., chemical desorption) is rate-

determining, we have:

H3O+ + e- H (ad) + H2O k1, k-1 (19.140)

H (ad) + H (ad) H2 k2 (19.141)

Here, the first step (Equation 19.140) is at quasi-equilibrium. Thus:

k1,0[H3O+](1 - H)exp[-(1 - )FE/RT] = k-1,0Hexp[FE/RT] (19.142)

That is,

H = {k1,0[H3O+]exp[-FE/RT]}/{k-1,0 + k1,0[H3O+]exp[-FE/RT]} (19.143)

For low overpotentials, k1,0[H3O+]exp[-FE/RT] << k-1,0, and Equation 19.143 reduces to:

H = (k1,0/k-1,0)[H3O+]exp[-FE/RT]} (19.144)

The rate equation for the second step (Equation 19.141) is given by:

i2 = Fk2H2 (19.145)

Thus, it follows from Equations 19.144 and 19.145 that the net current is:

i = 2i2 = 2Fk2H2 = 2Fk2 (k1,0/k-1,0)2[H3O+]2exp[-2FE/RT]} (19.146)

The electrochemical-desorption mechanism. For case (a), where the first step is rate-

determining, we have:

H3O+ + e- H (ad) + H2O k1 (19.147)

H (ad) + H3O+ + e- H2 + H2O k2, k-2 (19.148)

It can be seen that Equation 19.147 is the same as Equation 19.135, i.e., it represents the same

situation as that treated above for the chemical-desorption mechanism. Thus the same net rate

law is obtained (Equation 19.139).

On the other hand, for case (b), where the electrochemical desorption step is rate-

determining, the reaction steps can be represented as:

H3O+ + e- H (ad) + H2O k1, k1 (19.149)

H (ad) + H3O+ + e- H2 + H2O k2 (19.150)

Here, the first step is the same as in case (b) of the chemical desorption pathway. Hence, we can

also use Equations 19.143 and 19.144 to represent H. For the second step, the relevant rate

equation is:

i2 = Fk2,0H[H3O+]exp[-(1 - )FE/RT]} (19.151)

Combining Equations 19.144 and 19.151 gives:

i2 = Fk2,0 (k1,0/k-1,0)[H3O+]2exp[-(2 - )FE/RT]} (19.152)

The net current is then given by:

i = 2i2 = 2Fk2,0 (k1,0/k-1,0)[H3O+]2exp[-(2 - )FE/RT]} (19.153)

A summary of kinetic parameters arising from the above reaction mechanisms is

presented in Table 19.5.

Table 19.5. Kinetic parameters for the hydrogen evolution reaction

Mechanism Rate-limiting step c H3O+ R'n order*

Chemical-desorption Discharge (1 - ) 1.0Desorption 2 2.0

Electrochemical- Discharge (1 - ) 1.0desorption Desorption (2 - ) 2.0

____________________________________________________________________________________________

* From (∂lni/∂ln[H3O+])E

19.3.4 The Oxygen Electrode

Pathways for Oxygen Reduction. Two overall pathways have been identified for oxygen

reduction in aqueous solutions. These are the four-electron pathway (Equations 19.154 and

19.155), and the two-electron pathway (Equations 19.156 and 19.157):

O2 + 4H+ + 4e- 2H2O (acid solutions) (19.154)

O2 + 2H2O + 4e- 4OH- (alkaline solutions) (19.155)

O2 + 2H+ + 2e- H2O2 (acid solutions) (19.156)

O2 + H2O + 2e- HO2- + OH- (alkaline solutions) (19.157)

It must be noted that the hydrogen peroxide produced in the two-electron pathway may undergo

further reaction:

H2O2 + 2H+ + 2e- 2H2O (acid solutions) (19.158)

2H2O2 2H2O + O2 (acid solutions) (19.159)

HO2- + H2O + 2e- 3OH- (alkaline solutions) (19.160)

2HO2- 2OH- + O2 (alkaline solutions) (19.161)

Table 19.10 Electrode materials and reaction pathways for oxygen reduction.

Reaction Pathway Electrode Material

Four-electron Platinum group metals

Other metals: Ag

Oxides: Pyrochlores (e.g., lead ruthenate)

Two-electron Graphite

Metals: Au, Hg, most oxide-covered metals

(e.g., Ni, Co)

Oxides: Most transition metal oxides

(e.g., NiO, spinels)

(Modified from E. Yeager, J. Mol. Catal., 38, 5 (1986)

M O

O

Pauling Model (End-on)

O

OM Griffiths Model (Side-on)

M

M

O

O

(Cis)

M

M

O

O

(Trans)

Bridge Model

Figure 19.12 Molecular orbital representations of the types of interactions of molecular oxygen with metal atoms.

The electrochemical reactions of oxygen are influenced by the nature of the metal

electrode. On the basis of the pathways noted above, electrode materials may be classified as

indicated in Table 19.10. The manner in which molecular oxygen first binds (i.e., adsorbs on) to

the metal atoms on the electrode surface determines whether the overall reduction reaction will

follow the four-electron or the two electron pathway. Molecular orbital representations of the

possible modes of attachment are illustrated in Figure 19.12. It can be seen from Figure 19.12

that with the Griffith and bridge-type interactions, both oxygens of O2 form M-O bonds. These

arrangements should favor the dissociation of O2 and therefore, the direct 4-electron process.

Oxygen Reduction on Bare Platinum in Acid Solution. The following rate mechanism has

been proposed:

O2 + H+ + e- HO2 (ads) k1 (19.162)

HO2 (ads) + H+ + e- O (ads) + H2O k2, k-2 (19.163)

O (ads) + H+ + e- OH (ads) k3, k-3 (19.164)

OH (ads) + H+ + e- H2O k4, k-4 (19.165)

The first step (Equation 19.162) is taken to be rate-determining, while the remaining

steps are assumed to be at quasi-equilibrium. Thus, assuming Temkin conditions, we can write,

in analogy with Equation 19.124,

i = -nFCA(1 - B)k'f,oexp[-(1-)nFE/RT]exp[-G*ads,/RT]

+ nFBk'r,oexp[nFE/RT]exp[-G*des,/RT] (19.124)

i1 = Fk'1[H+]pO2(1 - )exp[-(1 - )FE/RT] exp[-G*ads,/RT] (19.166)

where

G*ads, = G*ads,0 + r (19.122)

Inserting Equation 19.122 in Equation 19.166 gives:

i1 = Fk1[H+]pO2(1 - )exp[-(1 - )FE/RT] exp[-r/RT] (19.167)

The steps which follow the rate-determining step are assumed to be in quasi-equilibrium.

Thus, in the case of the last step,

Fk4[H+] exp[-(1 - )FE/RT] exp[(1 - )r/RT]

= Fk-4[H2O](1 - )exp[FE/RT] exp[-r/RT] (19.168)

Again, neglecting the pre-exponential terms in , Equation 19.168 reduces to:

Fk4[H+]exp[-(1 - )FE/RT] exp[(1 - )r/RT]

= Fk-4[H2O]exp[FE/RT] exp[-r/RT] (19.169)

It follows from Equation 19.169 that

exp[-r/RT] = K4[H+]exp[-FE/RT] (19.170)

Rearranging Equation 19.170 gives the surface coverage as:

= (F/r) + (2.3RT/r)pH -(2.3RT/r)log K4 (19.171)

Using Equation 19.170 in Equation 19.167 gives:

i1 = Fk1[H+]pO2exp[-(1 - )FE/RT] K4[H+]exp[-FE/RT] (19.171a)

i1 = Fk1K4[H+]pO2exp[-(1 - + )FE/RT] (19.171b)

For ≈ ≈ 1/2, Equation 19.171a becomes:

i1 = k[H+]pO2exp[-FE/RT] (19.172)

OH (ads) + H+ + e- H2O k4, k-4 (19.165)

Also, the relationship between and potential is approximately linear (see Equation 19.171):

= (RT/r)pH + (F/r)V (19.173)

Combining Equations 5, 6, and 7, it can be shown that:

i1 = Fk1[H+]3/2pO2exp[-FE/RT] (19.174)

With increase in the absolute value of the potential, becomes negligibly small and Langmuirian

behavior prevails, resulting in a change in the Tafel slope to -2RT/F.

The reaction on iridium has been rationalized in terms of the following mechanism:

O2 + 2M 2MO (Step 1) (19.175a)

MO + H+ + e- MOH (Step 2) (19.175b)

MOH + H+ + e- M + H2O (Step 3) (19.175c)

where Step 2 is rate-determining.

Oxygen Reduction on Bare Platinum in Alkaline Solution. Two pathways have been

identified, one of which involves H2O2 intermediate. The mechanism for the H2O2 intermediate

pathway is:

O2 + e- O2- (ads) k1 (19.176)

O2- (ads) + H2O O2

- + OH (ads) k2, k-2 (19.177)

OH (ads) e- H- k3, k-3 (19.178)

With step one rate-determining and Temkin adsorption for the peroxide intermediate, we get:

i1 = Fk1pO2(1 - )exp[-(1 - )FE/RT] exp[-(1 - )f()] (19.179)

where

f() = F/RT - ln[H+] (19.180)

For ≈ 0.5, we get the following rate equation:

i1 = Fk1[H+]1/2pO2exp[-FE/RT] (19.181)

When the cathodic potential is relatively high, 0 so that surface coverage no longer

follows Temkin behavior. Accordingly, the rate becomes:

i1 = Fk1pO2exp[-(1 - )FE/RT] (19.182)

A comparison of Equations 19.181 and 19.182 indicates a change in the Tafel slope from -RT/F

to -2RT/F as the surface coverage changes from Temkin to Langmuirian behavior.

Oxygen Reduction on Gold in Alkaline Solution. The following mechanism has been

proposed by Zurilla et al.:

O2 + e- O2- (ads) k1 (19.183)

2O2- (ads) + H2O O2 + HO2

- + OH- k2, k-2 (19.184)

The first step is believed to be rate-determining.

Oxygen Reduction on Iron in Neutral Solution. The following mechanism has been

proposed by Zurilla et al:

O2 O2 (ads) k1 (19.185)

O2 (ads) + H2O + e- O2H (ads) + OH- (19.186)

O2H (ads) + e- O2H- (19.187)

O2H- + H2O H2O2 + OH- (19.188)

On oxide-covered iron, the surface coverage of adsorbed intermediates (O2H (ads)) under

Temkin conditions is given by

= (F/r)E + (2.3RT/r) log[OH-] + const (19.189)

Thus with adsorption (Equation 19.183) as the rate-determining step, we can write the following

rate equation:

i1 = Fk'1pO2 exp[-(1 - )FE/RT] exp[-G*ads,/RT] (19.190)

where

G*ads, = G*ads,0 + r (19.191)

Using Equations 19.189 and 19.191 in 19.190,

i1 = Fk1pO2 [OH-]-1/2exp[-(1 - + )FE/RT] (19.192)

Since the rate-determining step is a chemical rather than an electrochemical reaction, (1 - ) = 0,

and assuming = 1/2, Equation 19.192 becomes:

i1 = Fk1pO2 [OH-]-1/2exp[-FE/2RT] (19.193)

On bare iron, the first electron transfer was found to be rate-determining:

O2 + e- O2- (ads) k1 (19.183)

Experimentally it was found that: ∂E/∂log i = -120mV, ∂log i/∂log pO2 = 1, ∂log i/∂pH = 0.

Oxygen Evolution on Platinum in Acid Solution. The oxygen evolution reaction may be

contrasted with the oxygen reduction reaction in that the oxidation reaction occurs on an oxide-

coated surface in contrast to the reduction reaction which may involve both oxide and bare

surfaces.

OH- OH + e- (19.194)

OH + OH- O- + H2O (19.195)

O- O + e- (19.196)

O + O O2 (19.197)

In the case of acid solutions at low potentials (low current densities),

H2O OH + H+ + e- (slow) (19.198)

OH + OH- O- + H2O (19.199)

O- O + e- (19.200)

O + O O2 (19.201)

At higher potentials, mediation by surface metal ions:

M(z-1)+ Mz+ + e- (19.202)

Mz+ + H2O M(z-1)+ + OH + H+ (slow) (19.203)

OH + OH- O- + H2O (19.204)

O- O + e- (19.205)

O + O O2 (19.206)

Alkaline solutions at low potentials (low current densities) (Conway, Langmuir, 1990),

OH- OH + e- (19.207a)

OH + OH- O- + H2O (19.207b)

O- O + e- (slow) (19.208)

O + O O2 (19.209)

19.3.5 The Hydrogen Peroxide Electrode

19.3.6 The Chlorine Electrode

19.4 Mixed Potentials

19.4.1 Multiple Redox Couples at an Electrode

Up to this point, we have confined our discussion of electrode kinetics to situations where

there is only a single overall electron transfer reaction at the electrode. There are many

important practical systems, however, where two or more overall electron transfer reactions

occur simultaneously at the same electrode. Examples are found in oxidative dissolution and in

electroless deposition.

Figure 19.9 Multiple electrode reactions and electron flow.

The concept of mixed potential is illustrated in Figures 19.9 and 19.10. Let us consider

the surface of a metallic particle immersed in a solution containing the reactants A and B.

Immediately after contact with the two reactants, the metal surface develops two sites, i.e.,

anodic and cathodic sites at different potentials E1 and E2 respectively. These different

potentials are a result of the anodic (Equation 19.261) and cathodic (reverse direction of

Equation 19.262) reactions, as shown in Figure 19.9. The presence of two different potentials on

the same surface represents a non-steady-state situation and in order to achieve a uniform surface

potential, electrons will begin to flow through the surface of the metal particle, from the anodic

site to the cathodic site.

n1e-

(19.261)

Dn2e-

(19.262)

e-

A C + n1e- B + n2e- D

E1 E2

k1

k-1

k2

k-2

The current flow reaches a steady-state when the net anodic and net cathodic currents

balance each other (i.e., are equal and opposite in sign and therefore the net current is zero). The

potential at this stage is termed the mixed potential, Em, as illustrated for various scenarios in

Figure 19.10. The term rest potential is often used. This term refers to the open circuit potential

irrespective of whether the observed potential is a mixed potential (i.e., due to multiple overall

electrode reactions, e.g., Equations 19.261 and 19.262) or a reversible potential (i.e., due to a

single overall electrode reaction, e.g., Equation 19.261). The resulting overall reaction is given

by:

n2A + n1B = n2C + n1D (19.263)

(a) Activation-controlled oxidation plus activation-controlled reduction

(b) Activation-controlled oxidation plus mass transfer-controlled reduction

(c) Mass transfer-controlled oxidation plus activation-controlled reduction

(d) Mass transfer-controlled oxidation plus mass transfer-controlled reduction

i

E E

E E

i

ii

ia

ic ic

icic

ia

iaia E 1,eq

E 2,eq

E 1,eqE 1,eq

E1,eq

E2,eq

E 2,eqE2,eq

EmixEmix

EmixEmix

Figure 19.10 Relationship between the steady-state reaction rate and the mixed potential.

It follows from Equations 19.261 and 19.262 that the corresponding equilibrium

potentials are given by:

E1,eq = E1o + (RT/n1F) ln (CC/CA) (19.264)

E2,eq = E2o + (RT/n2F) ln (CB/CD) (19.265)

19.4.2 Surface-Controlled Kinetics

Under conditions where both the overall anodic and the overall cathodic reactions are

activation-controlled, Figure 19.10a is the relevant current vs potential situation. Referring to the

two redox couples presented above (Equations 19.261 and 19.262), the corresponding reaction

rates can be expressed as:

r1 = dCC/dt = =dCA/dt = i1/n1F = k1CAa = k-1CC

c (19.266a)

r2 = dCB/dt = -dCD/dt = i2/n2F = k2CDd = k-2CB

b (19.266b)

It is assumed here that stirring is sufficiently high such that both the anodic and cathodic

reactions are completely activation controlled, i.e., CA = CAs, CB = CBs.

The stoichiometry of Equation 19.263 implies that at steady-state,

(1/n2)dCC/dt = -(1/n1)dCB/dt (19.267)

It follows then from Equations 19.266a, 19.266b, and 19.267 that

i1/n1n2F = -i2/n1n2F = r1/n2 = -r2/n1 = rcat (19.268)

Thus,

i1 = -i2 = im = r1n1F = -r2n2F = n1n2Frcat (19.269)

where im is the current density at the mixed potential (where E = Em).

When only the anodic component of the net current is significant for the first redox

couple (i.e., the A/C couple), we can rewrite Equation 19.266a as:

r1 = i1/n1F = k1CAa = k1, 0CA exp[1FEm/RT] (19.270a)

Similarly, when only the cathodic current predominates for the second redox couple (i.e., the

D/B couple), Equation 19.266b simplifies to:

r2 = i2/n2F = -k-2CBb = -k-2, 0CB

b exp[-(1 - 2)FEm/RT] (19.270b)

Recalling Equations 7 and 8, we can manipulate Equations 19.270a and 19.270b to derive

an expression for the mixed potential as:

1FEm/RT = -u1 ln(n1k1, 0CAa/n2k-2, 0CB

b) (19.271)

where

u1 = 1/(1 + 1 = 2) (19.272)

Using Equation 19.271 to substitute for Em in Equation 19.270a gives:

(19.273)

where

u2 = 1 – u1 (19.274)

It follows from Equation 19.268 and 19.273 that

rcat = r1/n2 = ( (19.275)

where

kcat = (19.276)

EXAMPLE 19.9 Surface-controlled colloidal gold catalysis of the ferricyanide-thiosulfate reaction

The overall reaction between ferricyanide (Feic) and thiosulfate (Thio) can be written as:

Fe(CN)63-

+ S2O32-

= Fe(CN)64-

+ 1/2S4O62-

(1)

This reaction was investigated by Freund and Spiro (J. Chem. Soc. Faraday Trans. 1, 82, 2277-2282 (1986)). Table E19.9 presents their results.

Table E19.9 Effects of reactant concentrations on the initial rates of the gold-sol-catalyzed Feic-Thio reaction

Concentration (10 -3 mol dm -3 ) Initial Rate (10 -9 mol dm -3 s -1 )

Feic K2SO4 Thio Na2SO4 (rexp)

1.0 0.0 10.0 - 47.5 0.8 0.3 10.0 - 39.8 0.6 0.6 10.0 - 38.2 0.4 0.9 10.0 - 27.1 0.2 1.2 10.0 - 19.5 0.1 1.35 10.0 - 14.2 1.0 - 10.0 0. 46.0 1.0 - 7.5 2.5 43.9 1.0 - 5.0 5.0 34.6 1.0 - 2.5 7.5 23.4 1.0 - 1.25 8.75 20.3 1.0 - 0.625 9.38 14.6 1.0 - 0.25 9.75 5.6

According to previous experiments, the electrochemical reaction order of ferricyanide in the cathodic process is unity and the corresponding order for thiosulfate in the anodic reaction is also unity. Show that the results presented in Table E19.9 are consistent with a completely surface-controlled mechanism for the Feic/Thio reaction in the presence of gold sols.

Solution

According to the electrochemical analysis presented in Section 19.4.2,

rcat = kcat (19.275)

Thus, comparing the overall reaction (Equation 1) and Equation 19.275, we note that Feic represents the oxidant (B) and Thio represents the reductant (A). Therefore, we can write:

rcat = kcat (CThio) (CFeic) (2)

We know from the problem statement that the electrochemical reaction order of ferricyanide in the cathodic process is unity and the corresponding order for thiosulfate in the anodic reaction is also unity. Therefore, Equation 2 becomes:

rcat = kcat(CThio) (CFeic) (3)

Figure E19.9a shows a plot of ln rexp vs ln CFeic. It is seen that a straight line of slope 0.53 is obtained by

least squares. A similar diagram is shown in Figure E19.9b for ln rexp vs ln CThio; in this case the slope is 0.42.

Therefore, the experimental data can be represented by the following rate law:

rexp = k(CThio)0.53(CFeic)0.42 (4)

Figure E19.9

Now, referring to Equation 19.274, we find that

u1 + u2 = 0.53 + 0.42 = 0.95 ~ 1 (5)

The fact that the sum of the experimentally based kinetic orders is nearly unity, shows that the experimental data conform to the requirement of Equation 19.274, and therefore the kinetic results are consistent with a complete surface controlled electrochemical mechanism.

____________________________________________________________________________________________

19.4.3 Mixed-Control Kinetics

If the assumption used above that C = Cs is relaxed, then the bulk and surface

concentrations of the reactants and the products can be related with the aid of Fick's first law.

Recalling the discussion in Section 19.2.5, for the reactants A and B we have:

NA = kdA(CA - CAs) (19.277a)

NB = kdB(CB – CBs) (19.277b)

Under steady-state conditions we must have

NA = r1 = i1/n1F (19.278a)

NB = r2 = i2/n2F (19.278b)

It follows from Equations 19.277a and 19.278a that

CAs = (1 – i1/iLA)CA (19.279)

where iLA is the limiting current density attained when CAs = 0:

iLA = n1FkdACA = kdACA (19.280)

Similarly, from Equations 19.277b and 19278b,

CBs = (1 = i2/iLB)CB (19.281)

where

iLB = n2FkdBCB = kdBCB (19.282)

Referring to Equations 19.270a and 19.270b, we recognize that we must now use surface

concentrations. Thus,

r1 = i1/n1F = k1CAsa (19.283a)

r2 = i2/n2F = -k-2CBsb (19.283b)

Replacing CAs and CBs with Equations 19.279 and 19.281 gives

r1 = i1/n1F = k1(1 – i1/iLA)aCAa (19.284a)

r2 = i2/n2F = -k-2(1 – i2/iLB)b CBb (19.284b)

Again, at the mixed potential, E – Em, and im = i1 = -i2. Therefore, it follows from

Equations 19.284a and 19.284b that:

ln im = u2 ln (n1k1) + au2 ln (1 – im/iLA) + au2 ln CA

+ u1 ln (n2k-2) + bu1 ln (1-im/iLB) + bu1 ln CB (19.285)

At infinite stirring speed, iL and, thus, Equation 19.285 becomes

ln im, = u2 ln (n1k1) + au2 ln CA + u1 ln (n2k-2) + bu1 ln CB (19.286)

Combining Equations 19.285 and 19.286 gives:

ln (im/im, ) = au2 ln (1 – im/iLA) + bu1 ln (1 – im/iLB) (19.287)

For conditions where the mixed current is smaller than the respective limiting currents,

i.e., im << iLA and im << iLB, Equation 19.287 simplifies to (see P19.15):

1/im = 1/im, + au2/iLA + bu1/iLB (19.288)

It can also be shown that when im << iLA and im << iLB, the mixed potential is given by (see

P19.16):

(1 + 1 - 2)FEm/RT = ln (CBb/CA

a) - ln (n1k1, 0/n2k-2, 0) + (a/iLA – b/iLB)im (19.289)

At infinite stirring speed, iL and Em Em, and Equation 19.289 becomes:

(1 + 1 - 2)FEm, /RT = ln (CBb/CA

a) - ln (n1k1, 0/n2k-2, 0) (19.290)

Substituting Equation 19.290 into Equation 19.289 gives the mixed potential as:

Em = Em, + [RT/F(1 + 1 - 2)](a/iLA – b/iLB)im (19.291)

Reaction at a rotating disc surface. Recall from Table 18.3 that the mass transfer coefficient (k i)

for a rotating disc is given by:

(kid/Di) = 0.62(d2/)1/2(/Di)1/3 (19.292)

That is,

ki = 0.62-1/6Di2/31/2 (19.293)

Recalling Equations 19.280 and 19.282, we can write the limiting currents as:

iLA= n1FkdACA = n1F(0.62-1/6DA2/3)CA1/2 = A1/2 (19.294a)

iLB = n2FkdBCB = n2F(0.62-1/6DB2/3)CB1/2 = B1/2 (19.294b)

Inserting Equations 19.294ª and 19.294b into Equation 19.288 gives:

1/im = 1/im, + (au2/A + bu1/B)-1/2 (19.295)

Alternatively, in view of Equation 19.269, we can write:

1/rcat = n2/r1 = -n1/r2 = n1n2F/im = 1/rcat, + n1n2F(au2/A + bu1/B)-1/2 (19.296)

The corresponding mixed potential is obtained from Equations 19.291, 19.294a, and 19.294b as:

Em = Em, + [RT/F(1 + 1 - 2)](a/A – b/B)im-1/2 (19.297a)

Alternatively, recalling Equation 19.269,

Em = Em, + [RTn1n2/(1 + 1 - 2)](a/A – b/B)rcat-1/2 (19.297b)

It can be shown that (see P19.X):

Em = Em, + (y/x)(1 – im/im, ) = Em, + (y/x)(1 – rcat/rcat, ) (19.298)

where x is the slope of the 1/rcat vs. -1/2 plot and y is the slope of the Em vs. rcat-1/2 plot.

19.4.4 Transport-Controlled Kinetics

When the rate of electron transfer is relatively fast while the rate of mass transfer is

relatively slow, the overall reaction will essentially proceed at equilibrium at the solid surface.

Under these conditions, the following relationship must hold:

E1s,eq = E2s,eq = Em (19.299)

where the subscript (s) denotes surface conditions.

Parallel to Equations 19.279 and 19.281, the surface concentrations of the products C and

D can be expressed as:

CCs = (1 + i1/iLC)CC (19.300a)

CDs = (1 + i2/iLD)CD (19.300b)

where

iLC = n1FkdCCC = kdCCC (19.301a)

iLD = n2FkdDCD = kdDCD (19.301b)

The positive signs in Equations 10.300a and 19.300b (cf negative signs in Equations 19.279 and

19.281) reflect the fact that the surface concentrations of C and D are higher than the respective

bulk concentrations. By using the conditions im = i1 = =i2, and Equations 19.279, 19.281,

19.300a, and 19.300b to relate the surface and bulk concentrations, while recalling Equations

19.264 and 19.265, we find:

E1s, eq = E1 + (RT/n1F) ln (CC/CA) + (RT/n1F) ln [(1 + im/iLC)/(1 - im/iLA)] (19.302a)

E2s, eq = E2 + (RT/n2F) ln (CB/CD) + (RT/n2F) ln [(1 - im/iLB)/(1 + im/iLD)] (19.302b)

When Equation 19.302a is subtracted from Equation 19.302b, we get upon

rearrangement,

(19.303)

where

(19.304)

Let define a new function g(im) as:

g(im) = (iLC) (iLD) f(im) (19.305)

Then from Equations 19.280, 19.282, 19.301a, 19.301b, and 19.305,

g(im) = f(im) (19.306)

Using Equation 19.303 to substitute for f(im) in Equation 19.306,

(19.307)

Initial Rate. If the reaction products C and D are absent from the initial reaction mixture, then at

the beginning of the reaction the following condition is satisfied:

im/iLC >> 1, im/iLD >> 1 (19.308)

Also, for t ,

im/iLA << 1, im/iLB << 1 (19.309)

Under these conditions it can be shown (see P19.17) that:

1/im = 1/W + n2/iLA + n1/iLB (19.310a)

1/rcat = n1n2F/W + n1n22/iLA + n1

2n2/iLB (19.310b)

and

W = (19.311)

It can also be shown (see P19.18) that the mixed potential is given by:

(F/RT)(n1 + n2)Em = (F/RT)(n1E1 + n2E2) + ln CB/CA + ln (kdD/kdC)

+ ln [(1 – im/iLB)/(1 – im/iLA)] (19.312)

EXAMPLE 19.10 The platinum-catalyzed hexacyanoferrate(III) plus iodide reaction

The reaction between Fe(CN)63- and I- is strongly catalyzed by platinum metal. The overall reaction is:

Fe(CN)63- + 3/2I- Fe(CN)6

4- + 1/2I3- (1)

The kinetics of this reaction was investigated by Freund and Spiro (J. Chem. Soc., Faraday Trans. 1, 79, 491-504 (1983)); the platinum catalyst was used in the form of a rotating disc.

(a) Experiments conducted to examine the effect of the disc stirring rate on the reaction rate gave the following results:

Rotation speed (rev min-1) 100 200 300 500 1000 2000

Reaction rate (10 -6 mol m -2 s -1 ) 2 2.75 3.65 4.43 6.65 9.65

Show that these results are consistent with a completely mass transfer-controlled process.(b) The potential of the disc was monitored in the course of the reaction and it was found to be

independent of the rotation speed. Rationalize this observation, assuming (as found in (a) above) a transport-controlled process.

(c) Presented below are experimental data which give the effects of the reactant concentrations on the

reaction rate (r). With the aid of these data, determine the reaction order for Fe(CN)63- and for I-.

[Feic] (10-3 mol dm-3) 0.2 0.5 1.0 2.0 5.0 1.0 1.0 1.0 1.0 1.0

[I-] (10-3 mol dm-3) 50 50 50 50 50 30 50 80 150 300

r (10-6 mol m-2 s-1) 1.27 2.59 4.43 6.64 12.2 2.59 4.43 6.57 10.4 10.4

Emix (mV) 279 286 295 299 307 307 294 282 265 244

(d) The experimental results presented above in part (c) also include data on the effects of the reactant concentrations on the mixed potential indicated by the platinum metal. Show that these trends are consistent with an electrochemical mechanism of the platinum catalysis.

Solution

(a) Figure E19.10a shows a plot of reaction rate, r, vs. stirring rate. Recalling Equation 19.311, we can write:

W = (2)

where

f = (3)

Also, in view of Equation 19.293, we can write:

kdC = kdC1/2 (4)

kdD = kdD1/2 (5)

From Equations 2, 4, and 5,

W = f1/2 (6)

where

f = (kdC) (kdD) f (7)

Also, recall Equation 19.310:

1/im = 1/W + n2/iLA + n1/iLB (19.310)

Using Equations 19.241a, 19.241b, and 6 to substitute for iLA, iLB, and 6 respectively in Equation 19.310 gives the mixed current density as:

im = m1/2 (8)

where

i/m = 1/f + n2/A + n1/B (9)

It follows from Equation 9 that a plot of im vs. 1/2 should give a straight line.It can be seen from Equation 8 that for a complete mass transport-controlled process, a

plot of the reaction rate vs. the square root of the disc rotation rate should give a straight line passing through zero. Figure E19.10a shows such a plot for the data provided. Thus, it is concluded that the experimental data do indeed follow complete mass transfer control.

Figure E19.10a

(b) The mixed potential that corresponds to total mass transfer control is given by Equation 19.312. Inspection of this equation reveals that it has no dependence on the speed of rotation. Therefore the experimental observations are consistent with the assumed total mass transfer control. Inserting Equations 19.294a, 19.294b, 3, 4, and 7 into Equation 19.312, we get:

(F/RT)(n1 + n2)Em = (F/RT)(n1E1 + n2E2) + ln (CB/CA) + ln (kdD/kdC)+ ln [(1 - m/B)/(1 - m/A)] (10)

It follows from Equation 10 that a plot of Em vs. w1/2 should have a zero slope.(c) Based on the overall reaction (Equation 1), the following half-reactions can be written:

3/2I- = 1/2I3- + e- (11)

Fe(CN)63- + e- = Fe(CN)6

4- (12)

Comparing Equations 1, 11, and 12 with Figure 19.9 and Equations 19.261, 19.262, and 19.263, it can be seen that the following correspondence exists: A:IF

-, B:Fe(CN)63- (feic), C:I3

-, D:Fe(CN)6

4- (feoc).Closer comparison of Equations 19.261 and 11 reveals, however, that whereas in the

former both A and C have the same value of the stoichiometric coefficient (i.e., 1), in the case of Equation 11, I3

- and I- have different stoichiometric coefficients, i.e., ½ and 3/2 respectively. In order to use the equations available in Section 9.4.4 properly, we must transform Equation 11 into a form in which both the reactant and product have the same stoichiometric coefficient. We proceed by taking 3I- to be equivalent to a hypothetical species I3

3-. Then we can write:

I33- = I3

- + 2e- (13)2Fe(CN)6

3- + 2e - = 2Fe(CN) 64- (14)

I33- + 2Fe(CN)6

3- = I3- + 2Fe(CN)6

4- (15)

It must be noted that Equation 15 is equivalent to:

3I- + 2Fe(CN)63- = I3

- + 2Fe(CN)64- (16)

where I33- in Equation 15 has been replaced with 3I -. Adopting Equation 15 as our working

Equation, it follows from Equation 19.263 that

n1 = 2, n2 = 1, = 1/(n1 + n2) = 1/3 (17)

It must be noted further, that the 3I-/I33- substitution involves both a stoichiometric and a

thermodynamic equivalence. In the case of the stoichiometric equivalence, we recognize that the concentrations are related as:

(18)

The thermodynamic equivalence can be derived by expressing the redox reaction for the I3-/I-

couple in the following two alternative ways:

I3- + 2e- = 3I- (19)

E1 = E1 + (RT/2F) ln ( ) (20)

I3- + 2e- = I3

3- (21)

E1 = E1 + (RT/2F) ln ( ) (22)

It follows from Equations 20 and 22 that:

= (23)

We now turn to Equation 19.310b:

1/rcat = n1n2F/W + n1n22/iLA + n1

2n2/iLB (19.310b)Bearing in mind Equation 17, as well as the correspondence A:I3

3-, B:Fe(CN)63- (feic), we can

write

1/rcat = (2)(1)F/W + (1/3)(2)(1)2F/iLA + (1/3)(1)(2)2F/iLB

= 2F/W + (2/3)F/iLA + (4/3)F/iLB (24)

Now, using Equations 19.280 and 19.282 to substitute for iLA and iLB, we get:

1/rcat = 2F/W + (2/3)F/kdACA + (4/3)F/kdBCB (25)

Referring to the stoichiometric equivalence (Equation 18),

CA = (26)

Using Equation 26 to substitute for CA in Equation 25 and noting that CB corresponds to CFeic, we can write:

1/rcat = 2F/W + 2F/kdACI- + (4/3)F/kdBCFeic (27)

Recalling Equation 19.311,

W = (19.311)

we can write,

W = (kdC)1/3(kdD)2/3CA1/3CFeic

2/3 exp[2F(E2 - E1)/3RT] (28)

We note again that CA refers to . However, now we recognize that the parameter W is

related to the thermodynamic treatment of the redox couples (see Equations 19.302a and 19.302b). Thus, in Equation 19.311 we must use the thermodynamic equivalence (Equation 23), i.e.,

CA = = (29)

Accordingly, Equation 19.311 becomes:

W = (kdC)1/3(kdD)2/3 CI-CFeic

2/3 exp[2F(E2 - E1)/3RT] (30)

or

W = fCI-CFeic

2/3 (31)

Combining Equations 27 and 31,

1/rcat = 2F/fCI-CFeic

2/3 + (4/3)F/kdBCFeic + 2F/kdACI- (32)

If the last term on the right hand side is relatively small, we can write:

q = [1/rcat – (4/3)F/kdBCFeic] = 2F/fCI-CFeic

2/3 (33)

Figure E19.a shows a plot of log q vs. log CFeic for constant CI-. The slope obtained, i.e., 0.66, is

consistent with the reaction order of 2/3 for CFeic in Equation 33. The corresponding plot of log q vs. log CI

- for constant CFeic is shown in Figure E19.b. The slope of 1.0 is also in consonance with the reaction order of unity for CI

- in Equation 33.(d) Referring to Equation 19.289, we can see that for constant CB,

dEm/d ln CA = =aRT/(1 + 1 - 2)F (34)

Similarly, for constant CA,

dEm/d ln CB = bRT/(1 + 1 - 2)F (35)

At the temperature of the experiments (5C), and assuming 1 = 2 = 0.5, we expect

dEm/d ln CA = -RT/F = -24.0 mV (36)

dEm/d ln CB = RT/3F = 7.99 mV (37)

Figure 19.yya shows a plot of Em vs. ln I- with a slope of -23.2 mV, consistent with theory (Equation 36). The corresponding Em vs. ln Feic plot, Figure 19.yyb, shows a slope of 8.2 mV also in agreement with theory (Equation 37).

19.5 Electron Transfer at Semiconductor Electrodes

19.5.1 Energy Levels in Electron and Hole Transfer

The kinetics of electrochemical reactions at the semiconductor/aqueous interface can be

considered in terms of the transfer of carriers (holes or electrons) from energy levels in the solid

to those in the aqueous phase, and vice-versa. In order to satisfy the principle of conservation of

energy, radiationless electron transfer is permissible only if the initial and final locations of the

electron are at the same energy level.

Consider that Equation 19.1 now refers to a reaction taking place at the

semiconductor/electrolyte interface. Then, as illustrated in Figure 19.14, two different reaction

paths are available, depending on whether electron transfer is with the conduction band or the

valence band. Electron transfer in the valence band can be visualized in terms of the transfer of

holes. Thus, we have an electron-transfer mechanism (Equation 19.312a), and a hole-transfer

mechanism (Equation 19.312b):

A+ + e- = B (19.312a)

A+ = B + h+ (19.312b)

It can be seen from Figure 19.14 that whether the electron or hole transfer pathway predominates

depends on the relative positions of the band edges of the semiconductor with respect to the

energy levels of the redox couple in the electrolyte.

VALENCE BAND

CONDUCTION BAND

Ev

Ec

h+

e-

A+ e- = B+

h+A+ = B +

SOLID AQUEOUS PHASE

Figure 19.14 Electron transfer at a semiconductor electrode.

In the case of the conduction band electron transfer (Equation 19.312), we can envisage

the electrode reaction thus:

Az+(aq) + (Occupied State) e- (s) B(aq) + (Vacant State) (s) (19.313)

The rate of electron transfer will be expected to be proportional to the density of empty states on

the aqueous species (A+) and the density of occupied energy states at the same energy level in

the solid phase. It will be recalled (Chapter 9) that the energy states in the semiconductor are

spread over a wide range of energies. Thus, consideration of all the energy levels will require an

integration to be performed. However, in general, only a few states are occupied in the

conduction band while most of the energy states in the valence band are occupied. Thus, as far as electron transfer is concerned, only the energy bands in the neighborhood of the band edges Ec

and Ev are relevant. Accordingly, for the conduction band process, we can express the rate as:

ic = ic+ + ic- (19.314a)

= kc+°(Ec)°CBs°WB(Ec)°Nc - kc-°(Ec)°CAs°WA(Ec)°ns (19.314b)

where kc+ and kc- are respectively the proportionality constants for the anodic and cathodic

conduction band electron reactions, (Ec) is the probability that the collision of an aqueous

species with the surface will result in electron transfer, CAs and CBs are the surface

concentrations of the oxidized and reduced aqueous species, WA(Ec) and WB(Ec) represent

respectively, the probability of finding the oxidized and the reduced species at an energy state that corresponds to Ec, Nc is the density of energy states in the conduction band, and ns is the

electron density at the surface.

Similarly, for a valence band process,

iv = iv+ + iv- (19.315a)

= kv+°(Ev)°CBs°WB(Ev)°ps - kv-°(Ev)°CAs°WA(Ev)°Nv (19.315b)

where kv+ and kv- are respectively the proportionality constants for the anodic and cathodic

valence band hole reactions, Nv is the density of energy states in the valence band, and ps is the

hole density at the surface.

19.5.2 Current-Potential Relations

At equilibrium, there is no net conduction band or valence band electron transfer. That

is,

iv = ic = 0 (19.316)

If the surface concentrations of electrons and holes at equilibrium are denoted by nseq and pseq

respectively, then it follows from Equations 19.314 - 19.316 that:

ic = ic,o [CBs/CB - (CAs/CA)(ns/nseq)] (19.317)

iv = iv,o [(CBs/CB)(ps/pseq) - (CAs/CA)] (19.318)

where CA and CB are respectively the bulk aqueous phase concentrations of the oxidized and

reduced species, while ic,o and iv,o represent respectively the exchange currents for the

conduction band and valence band processes and are given by:

ic,o = kc+°(Ec)°CB°WB(Ec)°Nc (19.319a)

= kc-°(Ec)°CA°WA(Ec)°nseq (19.319b)

iv,o = kv+°(Ev)°CB°WB(Ev)°pseq (19.320)

= kv-°(Ev)°CA°WA(Ev)°Nv (19.321)

Equations 19.317 and 19.318 take into consideration the fact that at equilibrium the bulk and surface concentrations of the aqueous reactants must be equal, i.e., CAs = CA and CBs = CB.

When there is no concentration gradient,

CAs/CA = CBs/CB = 1 (19.322)

Therefore Equations 19.317 and 19.318 become:

ic = ic,o [1 - (ns/nseq)] (19.323)

iv = iv,o [(ps/pseq) - 1)] (19.324)

Recall Equation 5.161,

ns = nb exp-e s/kT (19.325)

where nb is the bulk concentration of electrons in the conduction band of the semiconductor. It

follows that

ns/nseq = [nb exp (-e s/kT)]/[nb exp (-eseq/kT)] (19.326)

= exp (-sce/kT) (19.327)

where sc is the overpotential given by

sc = s - seq (19.328)

and seq is the equilibrium value of . Combination of Equations 19.323 and 19.327 gives:

i = ic,o [1 - exp (-sce/kT)] (19.329)

Also, recall that the bulk and surface hole concentrations are related thus:

ps = pb exp es/kT (19.330)

It follows from Equations 19.324 and 19.330 that:

iv = iv,o [exp (sce/kT) - 1)] (19.331)

A comparison of the Butler-Volmer equation (Equation 19.21) with Equation 19.329

indicates that for electron exchange with the conduction band, the apparent transfer coefficient is

zero. On the other hand, a similar comparison with the expression obtained for hole exchange

with the valence band (Equation 19.331) reveals that in this case the apparent transfer coefficient

is unity. Further examination of Equation 19.329 reveals that for a conduction band electron

transfer, the applied potential affects the cathodic current but not the anodic current. The reverse

situation occurs in the case of a valence band electron transfer. Here, as can be seen from

Equation 19.331, the anodic current is potential-dependent whereas the cathodic current has no

potential dependence.

log i

/(kT/e)

ivic

ic

iv

i v,o

ic,o

CATHODIC ANODIC

Figure 19.16 Current-voltage curves for conduction-band and valence-band electron transfers at the semiconductor/electrolyte interface (after Gerrischer, 1970, p.498).

Figure 19.16 shows a graphical representation of the current-voltage relations for

conduction band and valence band charge transfers, as described by Equations 19.329 and

19.331 respectively.

______________________________________________________________________________

EXAMPLE 19.15 Reduction of oxidizing agents on n-ZnO

(a) Ferricyanide ions were reduced at the n-ZnO electrode by applying a cathodic potential:

Fe(CN)+ e- = Fe(CN)

The cathodic current was found to vary in direct proportion to the surface concentration of electrons and to the ferricyanide concentration. Rationalize these observations in the light of the above discussion of conduction band electron transfer. (see Freund and Morrison, Surf. Sci., 9, 119 (1968); Morrison, Surface Sci., 15, 363 (1969))

(b) Figure E19.15a presents cathodic currents obtained from the reduction of a number of one-equivalent oxidizing agents at a ZnO electrode. Show that the observed results are consistent with a first-order dependence of the cathodic current on the surface electron density.(see van den Berghe et al., Surf. Sci., 39, 368 (1973)).

Solution

(a) Recall the expression derived above for the current-potential relation for a conduction band electron transfer:

i = ic,o [1 - exp (-sce/kT)] (19.329)

If the cathodic paartial reaction predominates, Equation 19.329 becomes:i = -ic,o exp (-sce/kT) (1)

where

ic,o = kc-°(Ec)°CA°WA(Ec)°nseq (19.319b)

= k°CA°nseq (2)

It can be seen from Equation 1 that the rate of the reduction reaction is proportional to the exchange current density, which, as can be seen from Equation 2 is proportional to CA and nseq .

Thus it follows Equations 1 and 2 that

i = -k°CA°nseq exp (-sce/kT) (3)

= -k°CA°ns (4)

where Equation 19.327 has been used to relate nseq and ns. Equation 4 shows that the reduction

rate is first order in the oxidant concentration and in the surface concentration of electrons, as was observed experimentally.

Figure E19.15a Cathodic currents for one-equivalent oxidizing agents at ZnO.

(b) From Equation 19.327 we know that ns is exponentially dependent on sce/kT. Thus it follows

from Equation 3 that

ln |i| = ln (k°CA°nseq ) - sce/kT (5)

The curves in Figure 19.15a give slopes of ~60mV/decade, as expected from theory.____________________________________________________________________________________________

EXAMPLE 19.16 Reduction of Fe3+ on CdS

Show that for the CdS/Fe3+ system a slope of 60 mV/decade is obtained from the experimental data. (see G, p.506, Fig. 24; also V. A. Tyagai, Elecktrokhimiya, 1, 377 (1965))______________________________________________________________________________

19.5.3 Transport of Charge Carriers in Semiconductors

Where a surface reaction involves the consumption of a minority carrier, the rate at which

this carrier is transported to the semiconductor electrolyte interface can determine the rate of the

overall reaction. It is therefore of interest to develop a quantitative description of carrier

diffusion in a semiconductor. In order to accomplish this, we must first determine the rate at

which holes and electrons are produced in the bulk of the semiconductor. Once the carriers have

been produced, they must then be transported to the solid/aqueous interface where the

electrochemical reaction takes place.

In the presence of current flow, the semiconductor electrode can be divided into three

regions: the bulk region, the diffusion region, and the space charge region, as illustrated in

Figure 19.17 for an n-type semiconductor (see Green, p.386). In this case, holes constitute the

minority carriers. In the bulk region, the equilibrium hole and electron distributions are

preserved. Between the bulk and space charge regions is the diffusion region, characterized by a

concentration gradient of holes. As a result of the hole-consuming surface reaction, the hole concentration at the edge of the space charge region (x=x1) will be lower than its equilibrium

value. A concentration gradient of holes therefore arises outside the space charge region.

Bulk Region Diffusion Region

Space Charge Region

po

p1

Jp

2x 0x1

x

Figure 19.17 The diffusion of holes in an n-type semiconductor

Recalling Equation 9.12a, it can be stated that, under steady-state conditions, the hole

concentration must satisfy the following differential equation:

Dpd2p/dx2 + rp = 0 (19.332)

where Dp is the diffusion coefficient of a hole, and rp is the net rate of hole production inside the

semiconductor. As noted previously, inside a semiconductor, electrons and holes can undergo

recombination. This process can be viewed in terms of a homogeneous reaction of the form:

e- + h+ = 0 (19.333)

The forward reaction represents recombination, while the reverse represents the generation of an

electron-hole pair. Assuming that Equation 19.333 represents an elementary reaction, the

corresponding rate equation can be expressed as:

r = dp/dt = dn/dt = rgen + rrec = kgen - krecnp

= krec (nopo -np) = krec (ni2 -np) (19.334)

Let us now consider an n-type semiconductor. Let us suppose that an excess of holes is

introduced by some means. Then the rate of recombination will be given by:

dp/dt = - krecnop (19.335)

In writing Equation 19.335, it has been assumed that since for an n-type semiconductor, electrons are the majority carriers, the approximation n=no can be made. It follows from

Equation 19.335 that,

ln(p/pinitial) = -krecnot (19.336)

We can define the lifetime of a minority carrier as the time it takes to decrease the concentration

of non-equilibrium holes or electrons to 1/e of its original value. Therefore, substituting (p/pinitial) = 1/e at t = tp in Equation 19.336 gives the lifetime of holes, tp as:

tp = 1/krecno (19.337)

The lifetime of the minority carrier (tp) can also be viewed as the average time excess minority

carriers remain in their excited state after generation (Morrison, p.11). It follows from Equations

19.334 and 19.337 that,

rp = dp/dt = (po - p)/tp (19.338)

Combining Equations 19.332 and 19.338, we get:

Dpd2p/dx2 + (po - p)/tp = 0 (19.339)

The relevant boundary conditions are:

x , p = po (19.340)

x = x1, p = p1 (19.341)

Applying these boundary conditions to Equation 19.339 gives:

(p - po) = (p1 - po) exp[(x1 - x)/Lp] (19.342)

where

Lp = (Dptp)1/2 (19.343)

The parameter Lp is termed the diffusion length. It represents the average distance the minority

carrier (a hole in this case) diffuses before recombination occurs. It follows from the definition of the diffusion length that any holes generated in the region between x=(x1+ Lp) and x1 will

enter the space charge region without recombining.

It follows from Equation 19.343 that

dp/dx = [(p1 - po)/Lp] exp[(x1 - x)/Lp] (19.344)

Therefore, for x=x1,

dp/dx = (p1 - po)/Lp (19.345)

The hole current at x=x1 is then given by:

ip = -eDpdp/dx (19.346a)

= -eDp(p1 - po)/Lp (19.346b)

When p1 = 0, a limiting current (ip,lim) is attained, i.e.,

ip,lim = eDppo/Lp (19.347)

It follows from Equations 19.46b and 19.347 that:

ip = -(p1 - po)ip,lim/po (19.348a)

or

(p1/po) = 1 - (ip/ip,lim) (19.348b)

With the aid of Equation 19.348b, we can obtain a relationship between the surface

concentration of holes and the space charge potential in the presence of current flow. The

surface reaction results in current flow between the bulk regions of the semiconductor and the

surface. This current flow results in a departure of the hole and electron distributions from their respective equilibrium values. That is, the condition np = n i

2 is no longer satisfied. If the

departure from equilibrium is not too drastic, the resulting situation can be viewed as a "quasi-

equilibrium". Under these conditions, the holes and electrons are not in mutual equilibrium.

However, it can be assumed that the holes (or electrons) at the surface are in equilibrium with the

holes (or electrons) at the edge of the space charge layer:

ps = p1exp(es/kT) (19.349)

ns = n1exp(-es/kT) (19.350)

Also, since the region outside the space charge consitutes an electroneutral region, n1 and p1 are

related to their equilibrium values, no and po as:

(n1 - n2) = (p1 -po) (19.351)

Recalling Equation 19.349, we can write:

s = (kT/e)ln(ps/p1) = (kT/e)ln[(ps/po)(po/p1)]

= (kT/e)ln(ps/po) - (kT/e)ln(p1/po) (19.352)

Combining Equations 19.348b and 19.352 gives:

s = (kT/e)ln(ps/po) - (kT/e)ln[1 - (ip/ip,lim)] (19.353)

We can define a hole-transport overvoltage, Tp, as

Tp = (kT/e)ln[1 - (ip/ip,lim)] (19.354)

The hole-transport overvoltage can be viewed as the contribution which hole transport makes to

the space-charge overvoltage. Using Equation 19.354, we can rewrite Equation 19.353 as:

s = (kT/e)ln(ps/po) - Tp (19.355)

At equilibrium, there is no net current flow and the corresponding space-charge potential

is then given by:

s,eq = (kT/e)ln(ps,eq/po) (19.356)

Therefore, with the aid of Equations 19.355 and 19.356, the space charge overvoltage can be

expressed as:

s = s - s,eq = (kT/e)ln(ps/po) - Tp - (kT/e)ln(ps,eq/po)

= (kT/e)ln(ps/ps,eq) - Tp (19.357)

Now, the total overvoltage is given by:

= s + H (19.358)

From 19.357 and 19.358,

(ps/ps,eq) = exp[e( - H + Tp)/kT] (19.359)

Alternatively, using Equation 19.354 to substitute for Tp in Equation 19.359, we get:

(ps/ps,eq) = [1 - (ip/ip,lim)] exp[e(-H)/kT] (19.360)

19.5.4 Redox Reactions

______________________________________________________________________________

FURTHER READING

1. R. A. Robinson and R.H. Stokes, Electrolyte Solutions, 2nd ed., Butterworths, London, 1959, pp. 41-48, 118-132, 284-335.

2. J. S. Newman, Electrochemical Systems, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1991.

3. E. L. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, New York, NY 1984.

4. J. O'M. Bockris and A.K.N. Reddy, Modern Electrochemistry, Plenum, New York, 1970, Vol. 2, pp. 287-460.

5. A. J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, pp. 86-135. (429-487?) (488-552).

6. A. K. Vijh, Electrochemistry of Metals and Semiconductors, Marcel Dekker, New York, NY, 1973, pp. 34-52.

7. S. R. Morrison, Electrochemistry at Semiconductor and Oxidized Metal Electrodes, Plenum, New York, 1980, pp. 79-113.

8. H. Gerischer, "Semiconductor Electrochemistry", in Physical Chemistry: An Advanced Treatise, Vol. 9a, H. Eyring, D. Henderson, and W. Jost, eds., Academic, New York, 1970, pp. 463-542.

9. K. J. Vetter, Electrochemical Kinetics, Academic, New York, 1967.

10. T. Erdey-Gruz, Kinetics of Electrode Processes, Wiley-Interscience, New York, 1972.

11. J. O'M. Bockris and S. U. M. Khan, Surface Electrochemistry, Plenum, New York, 1993.

12. J. Koryta, J. Dvorak, and L. Kavan, Principles of Electrochemistry, 2nd ed., Wiley, New York, 1993.

13. G. Prentice, Electrochemical Engineering Principles, Prentice Hall, Englewood Cliffs, NJ, 1991.

14. B. E. Conway, Theory and Principles of Electrode Processes, Roland Press, New York, 1965.

15. P. H. L. Notten, J. E. A. M. van den Meerakker, and J. J. Kelly, Etching of III-V Semiconductors. An Electrochemical Approach, Elsevier Advance Technology, Oxford, U.K., 1991.

16. C. M. A. Brett and A. M. O. Brett, Electrochemistry. Principles, Methods, and Applications, Oxford, New York, NY, 1993.

17. N. Sato, Electrochemistry at Metal and Semiconductor Electrodes, Elsevier, New York, NY, 1998.

SECTION 19.3.5

1. J. O'M. Bockris and A.K.N. Reddy, Modern Electrochemistry, Plenum, New York, 1970, Vol. 2, pp. 1231-1264.

SECTION 19.4

1. M. Spiro, "Heterogeneous Catalysis in Solution. Part 17. Kinetics of Oxidation-Reduction Rections Catalyzed by Electron Transfer through the Solid: An Electrochemical Treatment", J. Chem. Soc., Faraday Trans. 1, 75, 1507-1512 (1979).

2. P. L. Freund and M. Spiro, "Heterogeneous Catalysis in Solution. Part 22. Oxidation-Reduction Rections Catalyzed by Electron Transfer through the Solid: Theory for Partial and Complete Mass-transport Control", J. Chem. Soc., Faraday Trans. 1, 79, 471-490 (1983).

3. P. L. Freund and M. Spiro, "Heterogeneous Catalysis in Solution. Part 23. Kinetics of a Redox system Showing Complete Mass-transport Control: The Hexacyanoferrate (III) + Iodide Reaction at a Rotating-platinum-disc Catalyst", J. Chem. Soc., Faraday Trans. 1, 79, 491-504 (1983).

4. D. S. Miller, A. J. Bard, G. McLendon, and J. Ferguson, “Catalytic Water Reduction at Colloidal Metal ‘Microelectrodes’. 2. Theory and Experiment”, J. Am. Chem. Soc., 103, 5336-5341 (1981).

SECTION 19.6

1. H. Gerischer, "Semiconductor Electrode Reactions", in Adv. Electrochem. Electrochem. Eng., P. Delahay, ed., 1, 139-232 (1961).

2. H. Gerischer, "Semiconductor Electrochemistry", in Physical Chemistry: An Advanced Treatise, Vol. 9a, H. Eyring, D. Henderson, and W. Jost, eds., Academic, New York, 1970, pp. 463-542.

3. A. K. Vijh, Electrochemistry of Metals and Semiconductors, Marcel Dekker, New York, NY, 1973, pp. 34-52.

4. S. R. Morrison, Electrochemistry at Semiconductor and Oxidized Metal Electrodes, Plenum, New York, 1980, pp. 79-113.

5. R. Memming, "Processes at Semiconductor Electrodes", in Comprehensive Treatise of Electrochemistry, Vol. 7, B. E. Conway, J. O'M. Bockris, E. Yeager, S. U. M. Khan, and R. E. White, eds., Plenum, New York, 1983, pp.529-592.

6. Yu. V. Pleskov anf Yu. Ya. Gurevich, Semiconductor Photoelectrochemistry, Consultants Bureau, New York, 1986.

7. H. Gerischer, "The Impact of Semiconductors on the Concepts of Electrochemistry", Electrochim. Acta, 35, 1677-1699.

8. N. Sato, Electrochemistry at Metal and Semiconductor Electrodes, Elsevier, New York, NY, 1998.