CHAPTER 19ELECTROCHEMICAL KINETICS
19.1Introduction 1
19.2Electron Transfer at Metal Electrodes 3
19.2.1Electron Transfer and Reaction Rate 3
19.2.2Activation Energy for Electron Transfer 5
19.2.3The Butler-Volmer Equation 8
19.2.4The Tafel Equations 13
19.2.5Mass Transfer Effects 15
19.2.6Electrical Double Layer Effects 19
19.3Redox Electrodes 20
19.3.1Multi-step Reactions at Metal Electrodes 20
19.3.2Adsorbed Intermediates and Electron Transfer 25
19.3.3The Hydrogen Electrode 29
19.3.4The Oxygen Electrode 33
19.3.5The Hydrogen Peroxide Electrode 41
19.3.6The Chlorine Electrode 41
19.4Mixed Potentials 41
19.4.1Multiple Redox Couples at an Electrode 41
19.4.2Surface-Controlled Kinetics 43
19.4.3Mixed-Control Kinetics 47
19.4.4Transport Controlled Kinetics 50
19.5Electron Transfer at Semiconductor Electrodes 57
19.5.1Energy Levels in Electron and Hole Transfer 57
19.5.2Current-Potential Relations 59
19.5.3Transport of Charge Carriers in Semiconductors 63
19.5.4Redox Reactions 68
______
19.1Introduction
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.
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.
19.2Electron Transfer at Metal Electrodes
19.2.1Electron 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)
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.
19.2.2Activation 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., kf 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.
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)
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:
(19.11a)
(19.11b)
It follows from Equations 19.10 a, c and 19.11 a, b, that
(19.12a)
(19.12b)
where
(19.13a)
(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)
19.2.3The Butler-Volmer Equation
At equilibrium, E = Eeq. Also, at equilibrium, rA = 0, i=0, and therefore (recalling Equations 19.7 and 19.8),
(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:
(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
(19.18a)
(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
(1)
Solution
From Equation 19.8 we know that
(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
(2)
Thus,
(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:
(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:
(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:
(19.23a)
(19.23b)
From 19.21a and 19.23a,
(19.24a)
From 19.21b and 19.23b,
(19.24b)
Using Equations 19.24a and 19.24b in Equation 19.20,
= 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):
(1)
That is,
(2)
Multiply Equation 2 by 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:
(19.27)
Similarly, from 19.8 and 19.20,
i+ = nFkr CB(19.28)
Compare 19.24b and 19.28:
(19.29)
Summary
(19.11a)
(19.12a)
(19.18a)
(19.27)
(19.11b)
(19.12b)
(19.18b)
(19.29)
19.2.4The 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 ||.
______
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)
______
19.2.5Mass 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)
Rate of consumption of A by the reaction at the electrode:
(19.32c)
Under steady-state conditions, we must have
(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,
(19.36)
Recalling Equation 19.33, NA = NB = rA, and thus, Equation 19.36 becomes:
(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:
(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:
(19.39)
Writing rAin 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:
(19.40a)
(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, kf » kr . Therefore, using the constraints iLa = 0, and kf » kr, we get from Equation 19.40b:
(19.41a)
Rearranging,
(19.41b)
Recalling Equation 19.38, (i.e., iLC = - nFkdACA), Equation 19.41b becomes
(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:
(19.41d)
or
(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:
(19.41f)
or
(19.41g)
In this case the reaction is transport controlled.
CASE 2: Only B is present in solution. HereCA = 0 iLC = 0. Also kr » kf (for reverse of reasons given above for Case 1). Therefore, Equation 19.40b simplifies to:
(19.42a)
or
(19.42b)
Recalling Equation 19.39 (i.e., iLa = nFkdBCB), Equation 19.42b may be rearranged as:
(19.42c)
When kdB » kr, the reaction is under kinetic control:
(19.42d)
i.e.,
(19.42e)
On the other hand, when kdB « kr, the reaction is under transport control and
(19.42f)
(19.42g)
19.2.6Electrical Double Layer Effects
19.3Redox Electrodes
19.3.1Multi-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.
19.3.2Adsorbed 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)