- Email: [email protected]
ELECTROCHEMICAL THEORY | Electrokinetics
Electrokinetics S Trasatti, University of Milan, Milan, Italy & 2009 Elsevier B.V. All rights reserved.
Introduction Chemical systems are governed by three experimental variables (temperature, pressure, and composition), whereas electrochemical systems by four variables (temperature, pressure, composition, and electrical state). Thus, electrochemical systems enjoy an additional degree of freedom that makes them more versatile and easier to control. Chemical systems at equilibrium are unable to do or to receive work. They are energetically dead because the Gibbs free energy change DG ¼ 0. Electrochemical systems at equilibrium maintain the maximum potentiality to perform work intact because DG ¼ nFDE, where DE measures the electrical state of electrochemical systems, F is the Farady constant, and n is the number of electrons. This is one of the fundamental equations of electrochemical thermodynamics and expresses the possibility of converting electrical energy (nFDE) into chemical energy (DG) in reactors called electrolyzers, as well as chemical energy into electrical energy in reactors called electrochemical power sources. Two specific features characterize an electrochemical system: (i) It contains free charges (electrons and ions) and (ii) its structure consists of two electronic conductors in contact with an ionic conductor. Feature (i) is a necessary but not sufficient condition for a system to be electrochemical, whereas feature (ii) is a condition ‘sine qua non’. Electrochemical Systems The two electronic conductors in contact with the ionic conductor are called electrodes, whereas the ionic conductor is called electrolyte. Mostly, the former are metals and the latter is a solution of a salt that dissociates giving free ions (electrolyte solution). At equilibrium, electrochemical systems are characterized by DErev , which can be split between the two electrodes: DErev ¼ E1 E2
½1
where E is called electrode potential. Often, one of the electrodes is used just to measure the potential of the other, that is, it plays the role of a reference electrode: DE ¼ Erev Eref
½2
In this case, a third electrode is present in the solution (counterelectrode) to complete the electrochemical cell (reactor).
Electrochemical systems are heterogeneous systems because they consist of immiscible phases in contact. As DE or DG are moved from their equilibrium value, electrons and ions flow along the conductors to restore the equilibrium condition. Their flow is measured as an electric current in an external circuit connecting the two electrodes. At the metal/electrolyte phase boundary, there is a change in charge carriers from electrons to ions (and vice versa) and this unavoidably results in a chemical change (electrode reaction). Thus, the core of an electrochemical reactor (cell) is the metal/solution interfacial region (electrical double layer). In all other parts of electrochemical cells, the only process taking place is migration of charge carriers (or secondary chemical reactions).
Electrode Reactions Electrons or ions, depending on the nature of the electrode reaction, can cross the metal/solution interface. Thus, electron transfer and ion transfer reactions are to be distinguished. An example of the former is Fe3þ ðsolÞ þ e ðMÞ"Fe2þ ðsolÞ
½I
where M stands for metal and sol for solution. An example of the latter is Agþ ðsolÞ"Agþ ðMÞ
½II
In this case, ionic species cross the interface, thus passing from one phase (solution) to the other (solid) (and vice versa). If electrons are added to both sides of the equation, Agþ ðsolÞ þ e ðMÞ"AgðMÞ
½III
This is the customary way of writing such an electrode reaction, which, however, conceals the very microscopic event at the origin of the equilibrium. At Equilibrium: Exchange Current At equilibrium, an electrochemical reaction is characterized by an electrode potential Erev . Owing to the stochastic nature of charge transfer, the flux of events to the right equals statistically the flux of events to the left. It is possible to write -
v ¼’ v
½3
23
24
Electrochemical Theory | Electrokinetics
which, taking into account that electrochemical reactions involve charged species, becomes -
’
nF v ¼ I ¼ I ¼ nF’ v
½4
where n is the number of electrons flowing per molecular event and F is the amount of electricity of 1 mole of electrons (Faraday constant, FE96 500 C). v is expressed in mol s1, and I is the electric current expressed as C s1 (ampere A). Owing to the heterogeneous nature of electrochemical interfaces, the flow of charge species is better referred to the unit of surface area of the interface: j ¼
I S
½5
where j is the current density and S (usually) the apparent surface area. At Erev , the common value of the rate of electrochemical events across an electrode interface in opposite directions -
’
j ¼ j ¼ j0
½6
is called exchange current density. It measures the degree of reversibility of an electrode reaction. j0 cannot be measured directly because the current in the external circuit - -
j ¼jj¼0
½7
However, j0 can be obtained indirectly from experimental data (see later).
Overpotentials: Energy Dissipations Electrode reactions can be retarded by various factors, but these are three routinely occuring factors: 1. activation energies; 2. composition gradients across the interface; and 3. ohmic resistances.
Activation (Barrier) Overpotential (gact) Electron transfer between the Fermi level in a metal and energy levels of species in solution is per se radiationless, that is, not activated. Nevertheless, electrode reactions exhibit activation energies. These are related to fluctuations in the molecular configuration mainly of species in the solution (chemical activation energy). Because electrode reactions involve charged species, as an overpotential (Z) is applied, there is a shift in the energy level of charged species. The classical theory of Zact is based on the model of activated complex with quasi-parabolic energy curves of reactants and products that move vertically without changing shape as the energy of particles is changed by modifying the electrode potential (Figure 1). For a single-step reaction involving n electrons, the change in energy brought about by an overpotential (Z) is nFZ. The model leads to a linear variation of the activation energy with Z conventionally given by anFZ and (1 a)nFZ for the cathodic and anodic reactions, respectively. A given Z, depending on its sign, increases the
Ox + ne
Away from Equilibrium: Overpotential
Red
Z ¼ E Erev
½8
is called overpotential. Z measures the amount of electrical energy dissipated to overcome reaction resistances. It can be positive (anodic), thus favoring oxidation (anodic) reactions, or negative (cathodic), thus favoring reduction (cathodic) reactions. Conventionally, the current associated with oxidation (anodic) reactions is taken as positive. Because electrode reactions are conventionally written in the direction of reduction, for example, Agþ ðsolÞ þ e ðMÞ-AgðMÞ
½IV ’
it follows, according to such a convention, that j > 0 and ’ ’ jo0. Thus, if jj j > jj j, then j ¼ j j > 0.
-
Free energy
If E is moved from Erev , a net current j will flow across the electrochemical cell and in the external circuit. The quantity
ΔG0≠
ΔG≠
−(1−)nF
−nF
ΔG 0≠
ΔG≠
ΔG 0
Reaction coordinate
Figure 1 Energy curves for a generic electrode reaction. Subscript 0 is for equilibrium and DGa are activation energies. The application of a cathodic overpotential (Zo0) decreases ’ DGa while increasing DGa . a is the transfer coefficient ( symmetry factor for single-step reactions).
25
Electrochemical Theory | Electrokinetics
activation energy of the electrode reaction in a given direction, while decreasing the energy in the opposite direction. Here, the term a is the fractional number varying between 0 and 1 that measures the fraction of electrical energy transferred to the cathode activation energy. For this reason, a is called ‘transfer coefficient’. On the contrary, a can be defined also in terms of the ratio of the slopes of the energy curves at the intersection point, thus also called ‘symmetry factor’. It is evident that in a single-step reaction, transfer coefficient and symmetry factor coincide. In terms of classical theory of reaction rates, the rate of a cathodic reaction (current density of reduction) at a given Z is given by -
j ¼ const krev eðanF Z=RT Þ
j ¼ const krev eð1aÞnF Z=RT
j0 nF Z RT
½12
whose slope is called ‘charge transfer resistance’ because eqn [12] formally reproduces Ohm’s law. From this equation, provided n is known, j0 can be obtained. Tafel lines
In the high overpotential range as nF jZjcRT ðjZj > 50 mVÞ, one of the two terms in eqn [11] becomes negligible with respect to the other and the following equations are obtained: ’
j ¼ j0 eð1aÞnF Z=RT
½13
-
j ¼ j0 eðanF Z=RT Þ
½14
which are universally known, in semilogarithmic form, as Tafel lines (Figure 3): Z ¼ a þ b ln j
½10
½15
with (for a single-step reaction)
Thus, the net current density is given by h i j ¼ j0 eð1aÞnF Z=RT eðanF Z=RT Þ
j ¼
½9
where krev is the reaction constant at equilibrium (Erev). Equation [9] illustrates the additional degree of freedom that electrode reactions enjoy. On the contrary, for the same Z, the rate of the anodic reaction is ’
Near equilibrium, as nF jZj{RT ðjZjp5 mVÞ, eqn [11] approaches a linear form:
a¼
½11
Equation [11] is the classical Butler–Volmer equation, the fundamental equation of electrochemical kinetics (Figure 2).
RT ln j0 anF
½16
RT anF
½17
>0
b¼
ne
l li
I
e af
T
I In j0
I
Overpotential
I0
In j
I0
Ta
fe
l li
ne
<0
I
Figure 2 Current–overpotential curve from the Butler–Volmer ’ equation; I0 is the exchange current, I is the net current, I and I are the cathodic and anodic partial current, respectively.
Figure 3 The Butler–Volmer equation in semilogarithmic plot. Tafel lines become evident as nF ZcRT . A same intercept for the two Tafel lines is a necessary (but not sufficient) condition for the extrapolated I0 to be considered the true exchange current.
Electrochemical Theory | Electrokinetics
Transfer coefficient (a)
The most probable value of a is 0.5, implying symmetric energy barrier or, alternatively, that the overpotential affects the activation energies for forward and backward reactions in opposite directions but to the same degree (Figure 4). Although other values are possible, the two limiting cases a ¼ 0 and a ¼ 1 are worth discussing in some detail. a ¼ 0 implies that further increase in Z does not produce further change in the (cathodic) activation energy. Under similar circumstances, an electrode reaction proceeds activationless, that is, there is no more activation energy for the given reaction. This situation can occur as the overpotential has been increased to such an extent that the whole activation energy has been balanced. In these conditions, the energy state of the products is at a much lower level than that of the reactant. Conversely, a ¼ 1 implies that further change in Z bears entirely and exclusively on the (cathodic) activation energy. Under similar circumstances, an electrode reaction proceeds barrierless, that is, there is no barrier for the given reaction. This situation can occur as the energy state of the products is at a much higher level than that of the reactants. It is evident that both a ¼ 0 and a ¼ 1 are extreme situations that can occur in a very restricted range of overpotential, the former at very high Z whereas the latter at very low Z. Also, as is obvious for the complementarity of a and (1 a), if the forward reaction is barrierless, the backward reaction is activationless, and vice versa. =0
= 0.5
=1
Concentration Overpotential As an electrode reaction proceeds, reactants are consumed while products are accumulated near the electrode surface. Thus, concentration profiles are formed that sustain diffusion fluxes of species toward or away from the electrode. Because mass transfer steps are in most cases slower than interfacial charge transfer steps (governed by the activation overpotential), the global rate of the electrode reaction will be limited by diffusion. Under similar circumstances, the concentration of reactants at the reaction site (ci , close to the electrode surface) will be lower than that in the bulk of the solution (ci) (Figure 5). Thus, to obtain a reaction rate commensurate to c, it is necessary to take the potential to a higher value (concentration overpotential) determined by the equation: Zconc ¼
RT ci ln nF ci
½18
In these conditions, the reaction rate equals the diffusion rate given by j ¼
nFDi ðci ci Þ d
½19
where Di is the diffusion coefficient of the reacting species and d the thickness of the diffusion layer. If the reaction rate is increased, a limit (jlim) is reached as ci ¼ 0, that is, reactant species are consumed as soon as they reach the reaction site. From eqn [19], because jlimpci and j pðci ci Þ, eqn [18] becomes Zconc ¼
RT jlim j ln nF jlim
½20
which allows us to calculate Zconc once jlim is experimentally known. If Zact and Zconc are both operative, eqn [20] allows us to correct the experimental points so that only Zact is
ci
1 2 Diffusion 3
0
Activationless
Normal
Barrierless
Figure 4 Evolution of the relative position of energy curves for reactants and products with variation of the transfer coefficient, a.
ci∗
Electrode
Where a and b are Tafel intercap and Tafel slope, respectively. Tafel lines possess a powerful diagnostic character for reaction mechanisms because the value of the slope acquires well-defined values for well-defined mechanisms. Such an option does not exist in chemical kinetics. Again, it is a consequence of the additional degree of freedom of electrochemical systems with respect to chemical systems. Besides, Tafel lines allow extrapolation of the linear section to Z ¼ 0 (Erev), with the intercept giving the exchange current density, j0.
Concentration
26
Distance
Figure 5 Sketch of the change in the stationary concentration profile of reactant i with variation of current. In the intermediate case, ci is the local concentration on the reaction plane and d is the thickness of the diffusion layer.
Electrochemical Theory | Electrokinetics
singled out and the Tafel line can emerge from the experimental data. Ohmic Resistances If ohmic resistances are present in the cell (e.g., poor conductivity of the electrode material, and/or finite distance between the electrode surface and the point where the reference electrode senses the electrode potential), ohmic drops (IR) add up a pseudo-overpotential term to the actual overpotential. Conceptually, IR is not an overpotential because it does not operate directly on the reaction rate (as either Zact or Zconc does). Technically, however, ohmic drops dissipate electrical energy so that the final outcome is the same.
1. In the plot of E versus ln j, a linear section can possibly be identified in the low-current-density region, and extrapolated to the high-current-density region. At constant current, the DE between the raw potential value and the extrapolated linear section is assumed to equal IR. If this is the case, a plot of DE versus j should result in a straight line passing through the origin whose slope gives R. If a curve results, this may indicate either a wrong estimate of DE or the presence of a broken Tafel line (two Tafel slopes). This is indicative of a change in reaction mechanism with overpotential. 2. In the presence of ohmic drops, the equation of the Tafel line becomes E ¼ a þ b ln I þ IR
Correction of the experimental current– overpotential curves
dE b ¼ þR dI I
Overpotential
½22
If (DE/DI) is plotted as a function of (1/I) from pairs of experimental values, a straight line with slope b (Tafel slope) and intercept R (ohmic resistance) is obtained. If two Tafel lines are present, a broken straight line results from which both values of b can be obtained. The actual value of R is that extrapolated from the linear section at higher current. Effect of Electrode Charge The concentration, overpotential Zconc arises as a consequence of dynamic concentration profiles. However, static concentration profiles in the electrode/solution boundary region can arise as a consequence of coulombic attraction or repulsion between the free charge on the electrode surface and charged species in solution. If this is the case, the concentration of reactants at the reaction site, ci , may differ from that in the bulk of the solution, ci : ci ¼ ci eðzi Ff
=RT Þ
½23
where f is the electric potential in the reaction plane. Not only ci but also the electrode potential must be reckoned in the reaction plane. Therefore, the reaction rate distortion is expressed as
conc
act
jexp ¼ jid eðanzÞF f
0
ln j L
½21
Differentiation with respect to I gives
The intrinsic overpotential of activation is the only one that cannot be avoided because it is related to the activation energies of the electrode reactions. Concentration overpotential and IR are rather experimental ‘accidents’ that can be minimized but not eliminated totally. However, they can possibly be corrected to extract only Zact (and therefore the Tafel line) from the raw experimental data. If Zconc and/or IR are operative besides Zact, experimental curves are nonlinear on an Z–log j plot (Figure 6). The presence of Zconc is easily verified by comparing the current at a given potential in the absence and in the presence of solution stirring. If stirring accelerates the reaction, this is indicative of Zconc. If jlim is obtained, Zconc can be corrected by using eqn [20]. If ohmic drops are such as to distort the Tafel line, corrections can be attempted instrumentally by the socalled current interruption method. If the instrumentation does not allow for that, two graphical approaches can be used; more subjective the former, more objective the latter.
ln j 0
27
ln j
Figure 6 Tafel line distorted by the effect of the concentration overpotential: Zact is the activation overpotential; Zcone is the concentration overpotential.
=RT
½24
which allows us to correct the raw experimental data for the so-called double-layer effect (also known as Frumkin effect) once f , a, n, and z are known. In the presence of double-layer effects, Tafel lines are distorted and the linearity is lost. Working in the presence of a large excess of background electrolyte can
28
Electrochemical Theory | Electrokinetics
alleviate the problem. For their very nature, double-layer effects vanish only at the potential of zero charge, where the charge on the electrode surface is zero. For the same reason, the major distortion of Tafel lines is observed around the potential of zero charge, while far away from Es ¼ 0, Tafel lines, although shifted from their ideal position, approach their ideal slope.
Multistep Reactions An electrode reaction involving more than one electron very unlikely proceeds in a single step. In most cases, n electrons are exchanged in n or more consecutive reactions whose sequence constitutes a reaction mechanism. At least n of the steps are electrochemical in nature (denoted by E), whereas others can be chemical in nature (denoted by C). Any sequence of steps (e.g., EE, EC, ECE) is a possible mechanism. Kinetic parameters, primarily the Tafel slope b, are sensitive to the sequence of steps. The mechanistic diagnostic power of b is unique and well expresses the additional degree of freedom of electrochemistry. In a multistep reaction DG 0 ¼ SDGi0 , where i is the ith step. But E a aSEia (activation energy). It is cusa tomarily assumed that E a Erds , where rds stands for rate-determining step, that is, the slowest step in the sequence. It ensues that, in multistep reactions, transfer coefficient and symmetry factor do not necessarily coincide. They do so only as the first electrochemical step is rate determining. The symmetry factor keeps its symbol a and coincides with the symmetry factor of the rds. Conversely, the transfer coefficient depends on the whole sequence of steps, thus acquiring more complex values. In practice, the transfer coefficient (denoted here by g, but such a symbol is not universally used) turns out to be a (very simple) function of a, g ¼ f (a). In addition, although g is experimentally accessible because it replaces a in the Tafel slope, a can only be estimated indirectly. For this reason, g is also called the observable transfer coefficient. Thus, although 0pap1, g can vary between wider limits (e.g., between 0 and 4 with n ¼ 4). A reaction mechanism is characterized by the stoichiometric number (denoted by n) that indicates the number of times the rds occurs in a complete act of reaction. Thus, in the same mechanism, different stoichiometric numbers are indicative of different rds. Stoichiometric numbers can be obtained either by comparing the exchange current obtained from the charge transfer resistance with the exchange current extrapolated from the Tafel line, or by comparing anodic and cathodic Tafel slopes in the assumption that the two Tafel lines refer to the same mechanism in opposite directions.
Generalized Tafel Slope Electrochemical kinetics is molecularly nonspecific, that is, kinetic parameters depend only on the details of the mechanism and not on the chemical nature of the species involved. Therefore, the way the Tafel slope depends on the reaction mechanism can be derived by analyzing a generic reaction R þ ne-P, proceeding in three steps involving n1, n2, and n3 electrons, and defined by the stoichiometric numbers n1, n2, and n3, respectively. If step 3 is assumed to be rate determining, the following Tafel slope is obtained: ðnÞrds RT dE ¼ b ¼ 2:303 d log j ðSnnÞords þ ðnnÞrds a F
½25
where subscript ‘ords’ stands for steps preceding the rds. The quantity in square brackets is the observable transfer coefficient. Equation [25] is derived on the assumption that all steps preceding the rds are at equilibrium, and that the coverage of the electrode surface with adsorbed intermediates is vanishingly small ðWi E0Þ. Equation [25] shows that Tafel slopes are all derived from (2.303RT/F), which at 25 1C takes the value 59 mV, approximated to 60 mV. It is possible to write b¼
60 mV g
½26
In case Wi E1, the assumption of equilibrium for the steps preceding the rds drops, and the kinetic analysis should be performed case by case. It turns out that g depends on the conditions of Wi for the same reaction mechanism and the same rds (see next section).
Reaction Mechanisms A mechanism is a sequence of steps, either electrochemical (where electrons are involved) denoted by E or chemical denoted by C. In the following analysis, each E step is assumed to be monoelectronic, the surface concentration of intermediates is negligible under Langmuirian conditions, and double-layer effects are nonexisting. Each mechanism for a given rds gives a well-defined Tafel slope (RT/gF), where g ¼ f (a) and a is assumed to equal 0.5. Tafel slopes are determined by the nature of the steps up to the rds only; the following steps proceed faster, and therefore, their nature does not affect the observed kinetic parameters. In the cases where the stoichiometric number can be determined, some information about the mechanism beyond the rds can be accessed because the stoichiometric number depends on the succession of the steps in the whole mechanism. The equation of Tafel lines can be obtained by writing down the expression of the reaction rate of the rds and
Electrochemical Theory | Electrokinetics
calculating the concentration of the intermediates from the equilibrium Nernst equation applied to the sum of the steps preceding the rds. As an example, let’s consider the following mechanism: ECEC. In case the first E is rate determining, the Tafel slope results (2.303RT/aF), that is, 120 mV (per decade of current). If the mechanism is EC*EC (where asterisk indicates the rds), b ¼ (2.303RT/F), that is, 60 mV. With ECE*C, b ¼ [2.303RT/ (1 þ a)F], that is, 40 mV. Finally, if the mechanism is ECEC*, b ¼ (2.303RT/2F), that is, 30 mV. Note that chemical steps determine the value of the Tafel slope only if they are rate determining. Otherwise, their presence is without effect on the kinetic parameters. Thus, ECE*C is kinetically equivalent to EE*. Similarly, ECEC* is kinetically equivalent to EEC*. Also, the symmetry factor appears explicitly in the expression of the Tafel slope only if an E step is rate determining. As a consequence, these mechanisms exhibit different Tafel slopes depending on the value of a, particularly in the cases of a ¼ 0 and a ¼ 1. If Wi E1, preceding steps can no longer be considered at equilibrium. As a consequence, the analysis leads to different values of Tafel slopes. As an example, for an EC* mechanism, the Tafel slope evolves from 60 mV to infinity, which is obvious because as the surface is saturated with intermediates, the reaction rate is governed by the rate of the chemical reaction and a variation of E can no longer have any further effect. Similarly, with an EE* mechanism, b evolves from 40 to 120 mV as Wi varies from 0 to 1. As a consequence, such a kind of mechanism can produce a broken Tafel line with two linear sections (Figure 7).
29
Broken Tafel lines with two linear sections can result also from a transition from one to another rds in the same mechanism. Thus, a transition from EE* to E*E would result in b changing from 40 to 120 mV, as for the case of Wi E0-1. In this case, additional criteria are necessary to distinguish the two situations, for example, direct determination of the surface concentration with intermediates. As a rule, mechanisms consist of consecutive steps. In this case, a change in rds with increasing overpotential leads always to an increase in Tafel slope. Occasionally, mechanisms may include steps in parallel. In this case, the reaction proceeds via the faster of the two steps. As a consequence, transition from one to another rds produces a decrease in the Tafel slope (Figure 8). Such a feature constitutes clear experimental evidence for a mechanism with parallel steps.
Other Kinetic Parameters Although the Tafel slope is a uniquely electrochemical kinetic parameter, reaction order (ni) and activation energy (Ea) are typical of chemical kinetics. If determined for electrode reactions, they can help establish the electrode mechanism. Although the definition of ni and Ea does not differ for electrochemical reactions, an essential difference exists in that the additional variable of electrochemical systems must be kept constant. The latter can be either E or Z. Thus, two kinds of reaction orders can be defined and determined for electrode reactions: Eni at constant potential and Zni at constant overpotential. The former is chemically significant, whereas the latter is often fractional in that
qi˜ 1 Z ni
¼E ni 7g
½27
Electrode potential
where g is the observed transfer coefficient, and the ‘ þ ’ sign applies to reactants while the ‘ ’ sign to products.
(b)
Electrode potential
(a)
qi˜ 0
A
B
A
B ln(current)
Figure 7 Sketch to show the evolution of the Tafel line for mechanism EE* with transition of coverage with intermediate from yiE0 to yiE1.
ln(current)
ln(current)
Figure 8 Sketch to show the shape of a broken Tafel line in the case of (a) consecutive or (b) parallel steps A and B.
30
Electrochemical Theory | Electrokinetics
It follows that the recommended experimental arrangement is at constant E. If double-layer effects are operating, the distortion of the Tafel slope implies that the reaction order will be also affected ultimately. Fractional reaction orders are, as a rule, indicative of possible double-layer effects. These vanish only at the potential of zero charge, although the effects can be dramatic at large charge densities. Thus, a true value of ni ¼ 1 can vary between 0.5 and 2.5 depending on the sign of the electrode charge and the charge sign of the reacting particle. As for reaction orders, and also for activation energies, determination is possible either at constant E or at constant Z. In both cases, the Butler–Volmer model results in a linear dependence of Ea either on E or on Z, but the degree is different. Activation energies have no direct relation with reaction mechanisms, but in some cases they can supplement the information derived from other kinetic parameters, helping identify the nature of the reaction mechanism. Activation energies are also influenced by the electrode charge state. Distortion effects vanish only at the potential of zero charge, although they go through a maximum at intermediate values of surface charge density. These effects become negligible as the ionic strength of the solution is increased.
ci*
Nonclassical Theories
d DE DErev DG DG0 DGga DGa 0 c g gact gconc g ni /* !i
Experimental kinetic data are customarily processed by means of phenomenological equations such as the Butler–Volmer–Tafel approach. The experimental picture conforms mostly to the classical model. Deviations are sometimes interpreted as flaws in the original theory. In some cases, the validity of the classical model has been questioned openly. Modern theories based on molecular approaches and quantum models have been developed since the 1960s. These involve the names of Marcus (Nobel prize) for electron exchange reactions, Dogonadze, Kuznetsov, Koper, Schmickler, and others. The molecular description of the electrochemical elementary act of charge transfer is now in an advanced state of sophistication, whereas experimental data are still satisfactorily processed on the basis of the classical approach.
Nomenclature
Di E Eref Erev E ni Ea Er = 0 F I j jexp jid j lim j0 krev n R S T n ni zi a
concentration of reactants at the reaction site diffusion coefficient of species i electrode potential potential of the reference electrode electrode potential at equilibrium reaction order at constant E activation energy potential of zero charge Faraday constant electric current electric current density experimental electric current density electric current density corrected for double-layer effects limiting current density (mass transfer) exchange current density reaction constant at equilibrium number of electrons ohmic resistance surface area absolute temperature stoichiometric number reaction order with respect to species i charge number of species i transfer coefficient (symmetry factor) for single-step reaction thickness of the diffusion layer cell potential difference cell potential difference at equilibrium Gibbs free energy change standard Gibbs free energy change activation energy at overpotential Z activation energy at Z ¼ 0 observable transfer coefficient overpotential activation overpotential concentration (diffusion) overpotential reaction order at constant Z electric potential at the reaction plane surface coverage with species i
Abbreviations and Acronyms rds
rate-determining step
See also: Electrochemical Theory: Double Layer; Electrolytes: Overview; Electrochemical Theory: Kinetics; Thermodynamics.
Symbols and Units a b ci
Tafel intercept Tafel slope concentration of speciesi
Further Reading Bagotsky VS (2006) Fundamentals of Electrochemistry, 2nd edn. Hoboken, NJ: Wiley-Interscience.
Electrochemical Theory | Electrokinetics
Bard AJ and Faulkner LR (2000) Electrochemical Methods: Fundamentals and Applications. New York: Wiley. Bockris JO’M, Reddy AKN, and Gamboa-Aldeco M (1998) Modern Electrochemistry 2A: Fundamentals of Electrodics, 2nd edn. New York: Kluwer/Plenum. Diard J-P, Le Gorrec B, and Montella C (1996) Cine´tique E´lectrochimique. Paris: Hermann. Gileadi E (1993) Electrode Kinetics for Chemists, Chemical Engineering and Material Scientists. New York: VCH. Hamann CH, Hamnett A, and Vielstich W (2007) Electrochemistry, 2nd edn. Weinheim: Wiley-VCH.
31
Korita J, Dvorˇa´k J, and Kavan L (1993) Principles of Electrochemistry. Chichester: Wiley. Rieger PH (1987) Electrochemistry. Englewood Cliffs, NJ: Prentice-Hall. Sato N (1998) Electrochemistry at Metal and Semiconductor Electrodes. Amsterdam: Elsevier. Schmickler W (1996) Interfacial Electrochemistry. New York: Oxford University Press. Southampton Electrochemistry Group (2001) Instrumental Methods in Electrochemistry. Chichester: Horwood.
Introduction Chemical systems are governed by three experimental variables (temperature, pressure, and composition), whereas electrochemical systems by four variables (temperature, pressure, composition, and electrical state). Thus, electrochemical systems enjoy an additional degree of freedom that makes them more versatile and easier to control. Chemical systems at equilibrium are unable to do or to receive work. They are energetically dead because the Gibbs free energy change DG ¼ 0. Electrochemical systems at equilibrium maintain the maximum potentiality to perform work intact because DG ¼ nFDE, where DE measures the electrical state of electrochemical systems, F is the Farady constant, and n is the number of electrons. This is one of the fundamental equations of electrochemical thermodynamics and expresses the possibility of converting electrical energy (nFDE) into chemical energy (DG) in reactors called electrolyzers, as well as chemical energy into electrical energy in reactors called electrochemical power sources. Two specific features characterize an electrochemical system: (i) It contains free charges (electrons and ions) and (ii) its structure consists of two electronic conductors in contact with an ionic conductor. Feature (i) is a necessary but not sufficient condition for a system to be electrochemical, whereas feature (ii) is a condition ‘sine qua non’. Electrochemical Systems The two electronic conductors in contact with the ionic conductor are called electrodes, whereas the ionic conductor is called electrolyte. Mostly, the former are metals and the latter is a solution of a salt that dissociates giving free ions (electrolyte solution). At equilibrium, electrochemical systems are characterized by DErev , which can be split between the two electrodes: DErev ¼ E1 E2
½1
where E is called electrode potential. Often, one of the electrodes is used just to measure the potential of the other, that is, it plays the role of a reference electrode: DE ¼ Erev Eref
½2
In this case, a third electrode is present in the solution (counterelectrode) to complete the electrochemical cell (reactor).
Electrochemical systems are heterogeneous systems because they consist of immiscible phases in contact. As DE or DG are moved from their equilibrium value, electrons and ions flow along the conductors to restore the equilibrium condition. Their flow is measured as an electric current in an external circuit connecting the two electrodes. At the metal/electrolyte phase boundary, there is a change in charge carriers from electrons to ions (and vice versa) and this unavoidably results in a chemical change (electrode reaction). Thus, the core of an electrochemical reactor (cell) is the metal/solution interfacial region (electrical double layer). In all other parts of electrochemical cells, the only process taking place is migration of charge carriers (or secondary chemical reactions).
Electrode Reactions Electrons or ions, depending on the nature of the electrode reaction, can cross the metal/solution interface. Thus, electron transfer and ion transfer reactions are to be distinguished. An example of the former is Fe3þ ðsolÞ þ e ðMÞ"Fe2þ ðsolÞ
½I
where M stands for metal and sol for solution. An example of the latter is Agþ ðsolÞ"Agþ ðMÞ
½II
In this case, ionic species cross the interface, thus passing from one phase (solution) to the other (solid) (and vice versa). If electrons are added to both sides of the equation, Agþ ðsolÞ þ e ðMÞ"AgðMÞ
½III
This is the customary way of writing such an electrode reaction, which, however, conceals the very microscopic event at the origin of the equilibrium. At Equilibrium: Exchange Current At equilibrium, an electrochemical reaction is characterized by an electrode potential Erev . Owing to the stochastic nature of charge transfer, the flux of events to the right equals statistically the flux of events to the left. It is possible to write -
v ¼’ v
½3
23
24
Electrochemical Theory | Electrokinetics
which, taking into account that electrochemical reactions involve charged species, becomes -
’
nF v ¼ I ¼ I ¼ nF’ v
½4
where n is the number of electrons flowing per molecular event and F is the amount of electricity of 1 mole of electrons (Faraday constant, FE96 500 C). v is expressed in mol s1, and I is the electric current expressed as C s1 (ampere A). Owing to the heterogeneous nature of electrochemical interfaces, the flow of charge species is better referred to the unit of surface area of the interface: j ¼
I S
½5
where j is the current density and S (usually) the apparent surface area. At Erev , the common value of the rate of electrochemical events across an electrode interface in opposite directions -
’
j ¼ j ¼ j0
½6
is called exchange current density. It measures the degree of reversibility of an electrode reaction. j0 cannot be measured directly because the current in the external circuit - -
j ¼jj¼0
½7
However, j0 can be obtained indirectly from experimental data (see later).
Overpotentials: Energy Dissipations Electrode reactions can be retarded by various factors, but these are three routinely occuring factors: 1. activation energies; 2. composition gradients across the interface; and 3. ohmic resistances.
Activation (Barrier) Overpotential (gact) Electron transfer between the Fermi level in a metal and energy levels of species in solution is per se radiationless, that is, not activated. Nevertheless, electrode reactions exhibit activation energies. These are related to fluctuations in the molecular configuration mainly of species in the solution (chemical activation energy). Because electrode reactions involve charged species, as an overpotential (Z) is applied, there is a shift in the energy level of charged species. The classical theory of Zact is based on the model of activated complex with quasi-parabolic energy curves of reactants and products that move vertically without changing shape as the energy of particles is changed by modifying the electrode potential (Figure 1). For a single-step reaction involving n electrons, the change in energy brought about by an overpotential (Z) is nFZ. The model leads to a linear variation of the activation energy with Z conventionally given by anFZ and (1 a)nFZ for the cathodic and anodic reactions, respectively. A given Z, depending on its sign, increases the
Ox + ne
Away from Equilibrium: Overpotential
Red
Z ¼ E Erev
½8
is called overpotential. Z measures the amount of electrical energy dissipated to overcome reaction resistances. It can be positive (anodic), thus favoring oxidation (anodic) reactions, or negative (cathodic), thus favoring reduction (cathodic) reactions. Conventionally, the current associated with oxidation (anodic) reactions is taken as positive. Because electrode reactions are conventionally written in the direction of reduction, for example, Agþ ðsolÞ þ e ðMÞ-AgðMÞ
½IV ’
it follows, according to such a convention, that j > 0 and ’ ’ jo0. Thus, if jj j > jj j, then j ¼ j j > 0.
-
Free energy
If E is moved from Erev , a net current j will flow across the electrochemical cell and in the external circuit. The quantity
ΔG0≠
ΔG≠
−(1−)nF
−nF
ΔG 0≠
ΔG≠
ΔG 0
Reaction coordinate
Figure 1 Energy curves for a generic electrode reaction. Subscript 0 is for equilibrium and DGa are activation energies. The application of a cathodic overpotential (Zo0) decreases ’ DGa while increasing DGa . a is the transfer coefficient ( symmetry factor for single-step reactions).
25
Electrochemical Theory | Electrokinetics
activation energy of the electrode reaction in a given direction, while decreasing the energy in the opposite direction. Here, the term a is the fractional number varying between 0 and 1 that measures the fraction of electrical energy transferred to the cathode activation energy. For this reason, a is called ‘transfer coefficient’. On the contrary, a can be defined also in terms of the ratio of the slopes of the energy curves at the intersection point, thus also called ‘symmetry factor’. It is evident that in a single-step reaction, transfer coefficient and symmetry factor coincide. In terms of classical theory of reaction rates, the rate of a cathodic reaction (current density of reduction) at a given Z is given by -
j ¼ const krev eðanF Z=RT Þ
j ¼ const krev eð1aÞnF Z=RT
j0 nF Z RT
½12
whose slope is called ‘charge transfer resistance’ because eqn [12] formally reproduces Ohm’s law. From this equation, provided n is known, j0 can be obtained. Tafel lines
In the high overpotential range as nF jZjcRT ðjZj > 50 mVÞ, one of the two terms in eqn [11] becomes negligible with respect to the other and the following equations are obtained: ’
j ¼ j0 eð1aÞnF Z=RT
½13
-
j ¼ j0 eðanF Z=RT Þ
½14
which are universally known, in semilogarithmic form, as Tafel lines (Figure 3): Z ¼ a þ b ln j
½10
½15
with (for a single-step reaction)
Thus, the net current density is given by h i j ¼ j0 eð1aÞnF Z=RT eðanF Z=RT Þ
j ¼
½9
where krev is the reaction constant at equilibrium (Erev). Equation [9] illustrates the additional degree of freedom that electrode reactions enjoy. On the contrary, for the same Z, the rate of the anodic reaction is ’
Near equilibrium, as nF jZj{RT ðjZjp5 mVÞ, eqn [11] approaches a linear form:
a¼
½11
Equation [11] is the classical Butler–Volmer equation, the fundamental equation of electrochemical kinetics (Figure 2).
RT ln j0 anF
½16
RT anF
½17
>0
b¼
ne
l li
I
e af
T
I In j0
I
Overpotential
I0
In j
I0
Ta
fe
l li
ne
<0
I
Figure 2 Current–overpotential curve from the Butler–Volmer ’ equation; I0 is the exchange current, I is the net current, I and I are the cathodic and anodic partial current, respectively.
Figure 3 The Butler–Volmer equation in semilogarithmic plot. Tafel lines become evident as nF ZcRT . A same intercept for the two Tafel lines is a necessary (but not sufficient) condition for the extrapolated I0 to be considered the true exchange current.
Electrochemical Theory | Electrokinetics
Transfer coefficient (a)
The most probable value of a is 0.5, implying symmetric energy barrier or, alternatively, that the overpotential affects the activation energies for forward and backward reactions in opposite directions but to the same degree (Figure 4). Although other values are possible, the two limiting cases a ¼ 0 and a ¼ 1 are worth discussing in some detail. a ¼ 0 implies that further increase in Z does not produce further change in the (cathodic) activation energy. Under similar circumstances, an electrode reaction proceeds activationless, that is, there is no more activation energy for the given reaction. This situation can occur as the overpotential has been increased to such an extent that the whole activation energy has been balanced. In these conditions, the energy state of the products is at a much lower level than that of the reactant. Conversely, a ¼ 1 implies that further change in Z bears entirely and exclusively on the (cathodic) activation energy. Under similar circumstances, an electrode reaction proceeds barrierless, that is, there is no barrier for the given reaction. This situation can occur as the energy state of the products is at a much higher level than that of the reactants. It is evident that both a ¼ 0 and a ¼ 1 are extreme situations that can occur in a very restricted range of overpotential, the former at very high Z whereas the latter at very low Z. Also, as is obvious for the complementarity of a and (1 a), if the forward reaction is barrierless, the backward reaction is activationless, and vice versa. =0
= 0.5
=1
Concentration Overpotential As an electrode reaction proceeds, reactants are consumed while products are accumulated near the electrode surface. Thus, concentration profiles are formed that sustain diffusion fluxes of species toward or away from the electrode. Because mass transfer steps are in most cases slower than interfacial charge transfer steps (governed by the activation overpotential), the global rate of the electrode reaction will be limited by diffusion. Under similar circumstances, the concentration of reactants at the reaction site (ci , close to the electrode surface) will be lower than that in the bulk of the solution (ci) (Figure 5). Thus, to obtain a reaction rate commensurate to c, it is necessary to take the potential to a higher value (concentration overpotential) determined by the equation: Zconc ¼
RT ci ln nF ci
½18
In these conditions, the reaction rate equals the diffusion rate given by j ¼
nFDi ðci ci Þ d
½19
where Di is the diffusion coefficient of the reacting species and d the thickness of the diffusion layer. If the reaction rate is increased, a limit (jlim) is reached as ci ¼ 0, that is, reactant species are consumed as soon as they reach the reaction site. From eqn [19], because jlimpci and j pðci ci Þ, eqn [18] becomes Zconc ¼
RT jlim j ln nF jlim
½20
which allows us to calculate Zconc once jlim is experimentally known. If Zact and Zconc are both operative, eqn [20] allows us to correct the experimental points so that only Zact is
ci
1 2 Diffusion 3
0
Activationless
Normal
Barrierless
Figure 4 Evolution of the relative position of energy curves for reactants and products with variation of the transfer coefficient, a.
ci∗
Electrode
Where a and b are Tafel intercap and Tafel slope, respectively. Tafel lines possess a powerful diagnostic character for reaction mechanisms because the value of the slope acquires well-defined values for well-defined mechanisms. Such an option does not exist in chemical kinetics. Again, it is a consequence of the additional degree of freedom of electrochemical systems with respect to chemical systems. Besides, Tafel lines allow extrapolation of the linear section to Z ¼ 0 (Erev), with the intercept giving the exchange current density, j0.
Concentration
26
Distance
Figure 5 Sketch of the change in the stationary concentration profile of reactant i with variation of current. In the intermediate case, ci is the local concentration on the reaction plane and d is the thickness of the diffusion layer.
Electrochemical Theory | Electrokinetics
singled out and the Tafel line can emerge from the experimental data. Ohmic Resistances If ohmic resistances are present in the cell (e.g., poor conductivity of the electrode material, and/or finite distance between the electrode surface and the point where the reference electrode senses the electrode potential), ohmic drops (IR) add up a pseudo-overpotential term to the actual overpotential. Conceptually, IR is not an overpotential because it does not operate directly on the reaction rate (as either Zact or Zconc does). Technically, however, ohmic drops dissipate electrical energy so that the final outcome is the same.
1. In the plot of E versus ln j, a linear section can possibly be identified in the low-current-density region, and extrapolated to the high-current-density region. At constant current, the DE between the raw potential value and the extrapolated linear section is assumed to equal IR. If this is the case, a plot of DE versus j should result in a straight line passing through the origin whose slope gives R. If a curve results, this may indicate either a wrong estimate of DE or the presence of a broken Tafel line (two Tafel slopes). This is indicative of a change in reaction mechanism with overpotential. 2. In the presence of ohmic drops, the equation of the Tafel line becomes E ¼ a þ b ln I þ IR
Correction of the experimental current– overpotential curves
dE b ¼ þR dI I
Overpotential
½22
If (DE/DI) is plotted as a function of (1/I) from pairs of experimental values, a straight line with slope b (Tafel slope) and intercept R (ohmic resistance) is obtained. If two Tafel lines are present, a broken straight line results from which both values of b can be obtained. The actual value of R is that extrapolated from the linear section at higher current. Effect of Electrode Charge The concentration, overpotential Zconc arises as a consequence of dynamic concentration profiles. However, static concentration profiles in the electrode/solution boundary region can arise as a consequence of coulombic attraction or repulsion between the free charge on the electrode surface and charged species in solution. If this is the case, the concentration of reactants at the reaction site, ci , may differ from that in the bulk of the solution, ci : ci ¼ ci eðzi Ff
=RT Þ
½23
where f is the electric potential in the reaction plane. Not only ci but also the electrode potential must be reckoned in the reaction plane. Therefore, the reaction rate distortion is expressed as
conc
act
jexp ¼ jid eðanzÞF f
0
ln j L
½21
Differentiation with respect to I gives
The intrinsic overpotential of activation is the only one that cannot be avoided because it is related to the activation energies of the electrode reactions. Concentration overpotential and IR are rather experimental ‘accidents’ that can be minimized but not eliminated totally. However, they can possibly be corrected to extract only Zact (and therefore the Tafel line) from the raw experimental data. If Zconc and/or IR are operative besides Zact, experimental curves are nonlinear on an Z–log j plot (Figure 6). The presence of Zconc is easily verified by comparing the current at a given potential in the absence and in the presence of solution stirring. If stirring accelerates the reaction, this is indicative of Zconc. If jlim is obtained, Zconc can be corrected by using eqn [20]. If ohmic drops are such as to distort the Tafel line, corrections can be attempted instrumentally by the socalled current interruption method. If the instrumentation does not allow for that, two graphical approaches can be used; more subjective the former, more objective the latter.
ln j 0
27
ln j
Figure 6 Tafel line distorted by the effect of the concentration overpotential: Zact is the activation overpotential; Zcone is the concentration overpotential.
=RT
½24
which allows us to correct the raw experimental data for the so-called double-layer effect (also known as Frumkin effect) once f , a, n, and z are known. In the presence of double-layer effects, Tafel lines are distorted and the linearity is lost. Working in the presence of a large excess of background electrolyte can
28
Electrochemical Theory | Electrokinetics
alleviate the problem. For their very nature, double-layer effects vanish only at the potential of zero charge, where the charge on the electrode surface is zero. For the same reason, the major distortion of Tafel lines is observed around the potential of zero charge, while far away from Es ¼ 0, Tafel lines, although shifted from their ideal position, approach their ideal slope.
Multistep Reactions An electrode reaction involving more than one electron very unlikely proceeds in a single step. In most cases, n electrons are exchanged in n or more consecutive reactions whose sequence constitutes a reaction mechanism. At least n of the steps are electrochemical in nature (denoted by E), whereas others can be chemical in nature (denoted by C). Any sequence of steps (e.g., EE, EC, ECE) is a possible mechanism. Kinetic parameters, primarily the Tafel slope b, are sensitive to the sequence of steps. The mechanistic diagnostic power of b is unique and well expresses the additional degree of freedom of electrochemistry. In a multistep reaction DG 0 ¼ SDGi0 , where i is the ith step. But E a aSEia (activation energy). It is cusa tomarily assumed that E a Erds , where rds stands for rate-determining step, that is, the slowest step in the sequence. It ensues that, in multistep reactions, transfer coefficient and symmetry factor do not necessarily coincide. They do so only as the first electrochemical step is rate determining. The symmetry factor keeps its symbol a and coincides with the symmetry factor of the rds. Conversely, the transfer coefficient depends on the whole sequence of steps, thus acquiring more complex values. In practice, the transfer coefficient (denoted here by g, but such a symbol is not universally used) turns out to be a (very simple) function of a, g ¼ f (a). In addition, although g is experimentally accessible because it replaces a in the Tafel slope, a can only be estimated indirectly. For this reason, g is also called the observable transfer coefficient. Thus, although 0pap1, g can vary between wider limits (e.g., between 0 and 4 with n ¼ 4). A reaction mechanism is characterized by the stoichiometric number (denoted by n) that indicates the number of times the rds occurs in a complete act of reaction. Thus, in the same mechanism, different stoichiometric numbers are indicative of different rds. Stoichiometric numbers can be obtained either by comparing the exchange current obtained from the charge transfer resistance with the exchange current extrapolated from the Tafel line, or by comparing anodic and cathodic Tafel slopes in the assumption that the two Tafel lines refer to the same mechanism in opposite directions.
Generalized Tafel Slope Electrochemical kinetics is molecularly nonspecific, that is, kinetic parameters depend only on the details of the mechanism and not on the chemical nature of the species involved. Therefore, the way the Tafel slope depends on the reaction mechanism can be derived by analyzing a generic reaction R þ ne-P, proceeding in three steps involving n1, n2, and n3 electrons, and defined by the stoichiometric numbers n1, n2, and n3, respectively. If step 3 is assumed to be rate determining, the following Tafel slope is obtained: ðnÞrds RT dE ¼ b ¼ 2:303 d log j ðSnnÞords þ ðnnÞrds a F
½25
where subscript ‘ords’ stands for steps preceding the rds. The quantity in square brackets is the observable transfer coefficient. Equation [25] is derived on the assumption that all steps preceding the rds are at equilibrium, and that the coverage of the electrode surface with adsorbed intermediates is vanishingly small ðWi E0Þ. Equation [25] shows that Tafel slopes are all derived from (2.303RT/F), which at 25 1C takes the value 59 mV, approximated to 60 mV. It is possible to write b¼
60 mV g
½26
In case Wi E1, the assumption of equilibrium for the steps preceding the rds drops, and the kinetic analysis should be performed case by case. It turns out that g depends on the conditions of Wi for the same reaction mechanism and the same rds (see next section).
Reaction Mechanisms A mechanism is a sequence of steps, either electrochemical (where electrons are involved) denoted by E or chemical denoted by C. In the following analysis, each E step is assumed to be monoelectronic, the surface concentration of intermediates is negligible under Langmuirian conditions, and double-layer effects are nonexisting. Each mechanism for a given rds gives a well-defined Tafel slope (RT/gF), where g ¼ f (a) and a is assumed to equal 0.5. Tafel slopes are determined by the nature of the steps up to the rds only; the following steps proceed faster, and therefore, their nature does not affect the observed kinetic parameters. In the cases where the stoichiometric number can be determined, some information about the mechanism beyond the rds can be accessed because the stoichiometric number depends on the succession of the steps in the whole mechanism. The equation of Tafel lines can be obtained by writing down the expression of the reaction rate of the rds and
Electrochemical Theory | Electrokinetics
calculating the concentration of the intermediates from the equilibrium Nernst equation applied to the sum of the steps preceding the rds. As an example, let’s consider the following mechanism: ECEC. In case the first E is rate determining, the Tafel slope results (2.303RT/aF), that is, 120 mV (per decade of current). If the mechanism is EC*EC (where asterisk indicates the rds), b ¼ (2.303RT/F), that is, 60 mV. With ECE*C, b ¼ [2.303RT/ (1 þ a)F], that is, 40 mV. Finally, if the mechanism is ECEC*, b ¼ (2.303RT/2F), that is, 30 mV. Note that chemical steps determine the value of the Tafel slope only if they are rate determining. Otherwise, their presence is without effect on the kinetic parameters. Thus, ECE*C is kinetically equivalent to EE*. Similarly, ECEC* is kinetically equivalent to EEC*. Also, the symmetry factor appears explicitly in the expression of the Tafel slope only if an E step is rate determining. As a consequence, these mechanisms exhibit different Tafel slopes depending on the value of a, particularly in the cases of a ¼ 0 and a ¼ 1. If Wi E1, preceding steps can no longer be considered at equilibrium. As a consequence, the analysis leads to different values of Tafel slopes. As an example, for an EC* mechanism, the Tafel slope evolves from 60 mV to infinity, which is obvious because as the surface is saturated with intermediates, the reaction rate is governed by the rate of the chemical reaction and a variation of E can no longer have any further effect. Similarly, with an EE* mechanism, b evolves from 40 to 120 mV as Wi varies from 0 to 1. As a consequence, such a kind of mechanism can produce a broken Tafel line with two linear sections (Figure 7).
29
Broken Tafel lines with two linear sections can result also from a transition from one to another rds in the same mechanism. Thus, a transition from EE* to E*E would result in b changing from 40 to 120 mV, as for the case of Wi E0-1. In this case, additional criteria are necessary to distinguish the two situations, for example, direct determination of the surface concentration with intermediates. As a rule, mechanisms consist of consecutive steps. In this case, a change in rds with increasing overpotential leads always to an increase in Tafel slope. Occasionally, mechanisms may include steps in parallel. In this case, the reaction proceeds via the faster of the two steps. As a consequence, transition from one to another rds produces a decrease in the Tafel slope (Figure 8). Such a feature constitutes clear experimental evidence for a mechanism with parallel steps.
Other Kinetic Parameters Although the Tafel slope is a uniquely electrochemical kinetic parameter, reaction order (ni) and activation energy (Ea) are typical of chemical kinetics. If determined for electrode reactions, they can help establish the electrode mechanism. Although the definition of ni and Ea does not differ for electrochemical reactions, an essential difference exists in that the additional variable of electrochemical systems must be kept constant. The latter can be either E or Z. Thus, two kinds of reaction orders can be defined and determined for electrode reactions: Eni at constant potential and Zni at constant overpotential. The former is chemically significant, whereas the latter is often fractional in that
qi˜ 1 Z ni
¼E ni 7g
½27
Electrode potential
where g is the observed transfer coefficient, and the ‘ þ ’ sign applies to reactants while the ‘ ’ sign to products.
(b)
Electrode potential
(a)
qi˜ 0
A
B
A
B ln(current)
Figure 7 Sketch to show the evolution of the Tafel line for mechanism EE* with transition of coverage with intermediate from yiE0 to yiE1.
ln(current)
ln(current)
Figure 8 Sketch to show the shape of a broken Tafel line in the case of (a) consecutive or (b) parallel steps A and B.
30
Electrochemical Theory | Electrokinetics
It follows that the recommended experimental arrangement is at constant E. If double-layer effects are operating, the distortion of the Tafel slope implies that the reaction order will be also affected ultimately. Fractional reaction orders are, as a rule, indicative of possible double-layer effects. These vanish only at the potential of zero charge, although the effects can be dramatic at large charge densities. Thus, a true value of ni ¼ 1 can vary between 0.5 and 2.5 depending on the sign of the electrode charge and the charge sign of the reacting particle. As for reaction orders, and also for activation energies, determination is possible either at constant E or at constant Z. In both cases, the Butler–Volmer model results in a linear dependence of Ea either on E or on Z, but the degree is different. Activation energies have no direct relation with reaction mechanisms, but in some cases they can supplement the information derived from other kinetic parameters, helping identify the nature of the reaction mechanism. Activation energies are also influenced by the electrode charge state. Distortion effects vanish only at the potential of zero charge, although they go through a maximum at intermediate values of surface charge density. These effects become negligible as the ionic strength of the solution is increased.
ci*
Nonclassical Theories
d DE DErev DG DG0 DGga DGa 0 c g gact gconc g ni /* !i
Experimental kinetic data are customarily processed by means of phenomenological equations such as the Butler–Volmer–Tafel approach. The experimental picture conforms mostly to the classical model. Deviations are sometimes interpreted as flaws in the original theory. In some cases, the validity of the classical model has been questioned openly. Modern theories based on molecular approaches and quantum models have been developed since the 1960s. These involve the names of Marcus (Nobel prize) for electron exchange reactions, Dogonadze, Kuznetsov, Koper, Schmickler, and others. The molecular description of the electrochemical elementary act of charge transfer is now in an advanced state of sophistication, whereas experimental data are still satisfactorily processed on the basis of the classical approach.
Nomenclature
Di E Eref Erev E ni Ea Er = 0 F I j jexp jid j lim j0 krev n R S T n ni zi a
concentration of reactants at the reaction site diffusion coefficient of species i electrode potential potential of the reference electrode electrode potential at equilibrium reaction order at constant E activation energy potential of zero charge Faraday constant electric current electric current density experimental electric current density electric current density corrected for double-layer effects limiting current density (mass transfer) exchange current density reaction constant at equilibrium number of electrons ohmic resistance surface area absolute temperature stoichiometric number reaction order with respect to species i charge number of species i transfer coefficient (symmetry factor) for single-step reaction thickness of the diffusion layer cell potential difference cell potential difference at equilibrium Gibbs free energy change standard Gibbs free energy change activation energy at overpotential Z activation energy at Z ¼ 0 observable transfer coefficient overpotential activation overpotential concentration (diffusion) overpotential reaction order at constant Z electric potential at the reaction plane surface coverage with species i
Abbreviations and Acronyms rds
rate-determining step
See also: Electrochemical Theory: Double Layer; Electrolytes: Overview; Electrochemical Theory: Kinetics; Thermodynamics.
Symbols and Units a b ci
Tafel intercept Tafel slope concentration of speciesi
Further Reading Bagotsky VS (2006) Fundamentals of Electrochemistry, 2nd edn. Hoboken, NJ: Wiley-Interscience.
Electrochemical Theory | Electrokinetics
Bard AJ and Faulkner LR (2000) Electrochemical Methods: Fundamentals and Applications. New York: Wiley. Bockris JO’M, Reddy AKN, and Gamboa-Aldeco M (1998) Modern Electrochemistry 2A: Fundamentals of Electrodics, 2nd edn. New York: Kluwer/Plenum. Diard J-P, Le Gorrec B, and Montella C (1996) Cine´tique E´lectrochimique. Paris: Hermann. Gileadi E (1993) Electrode Kinetics for Chemists, Chemical Engineering and Material Scientists. New York: VCH. Hamann CH, Hamnett A, and Vielstich W (2007) Electrochemistry, 2nd edn. Weinheim: Wiley-VCH.
31
Korita J, Dvorˇa´k J, and Kavan L (1993) Principles of Electrochemistry. Chichester: Wiley. Rieger PH (1987) Electrochemistry. Englewood Cliffs, NJ: Prentice-Hall. Sato N (1998) Electrochemistry at Metal and Semiconductor Electrodes. Amsterdam: Elsevier. Schmickler W (1996) Interfacial Electrochemistry. New York: Oxford University Press. Southampton Electrochemistry Group (2001) Instrumental Methods in Electrochemistry. Chichester: Horwood.