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J.A.V. Butler Max Volmer Walter Nernst Julius Tafel Physical and Interfacial Electrochemistry 2013 Lecture 6. Phenomenological electrode kinetics: The Butler-Volmer Equation. 1 F F i i0 exp exp RT RT iox ired Kinetics of interfacial ET. • Estimation of equilibrium redox potentials provides a quantitative measure for the tendency for a specific redox reaction to occur. Kinetic information is not derived. • ET reactions at electrode/solution interfaces are activated processes. – ET rate effected by: • Applied electrode potential • Temperature – Activation energy barrier height can be effected by applied potential. This is in contrast to ordinary chemical reactions. • We seek an answer to the following questions: – How can we quantitatively model the rate of an ET process which occurs at the interface between a metallic electrode and an aqueous solution containing a redox active couple? – How can kinetic information about ET processes be derived? • We shall also investigate the influence of material transport, and double layer structure on interfacial ET processes. Basic concepts of electrode kinetics. • For an interfacial ET process: – current flow is proportional to reaction flux (rate). • Reaction rate is proportional to reactant concentration at interface. • As in chemical kinetics: – the constant of proportionality between reaction rate fS (molcm2s-1) and reactant concentration c (molcm-3) is termed the rate constant k (cms-1). • All chemical and electrochemical reactions are activated processes. • An activation energy barrier exists which must be overcome in order that the chemical reaction may proceed. – Energy must be supplied to surmount the activation energy barrier. – This energy may be supplied thermally or also (for ET processes at electrodes) via the application of a potential to the metallic electrode. The electrode potential drives ET processes at interfaces. • Application of a potential to an electrode generates a large electric field at the electrode/solution interface which reduces the height of the activation energy barrier and thereby increases the rate of the ET reaction. • Hence the applied potential acts as a driving force for the ET reaction. • We intuitively expect that the current should increase with increasing driving force. This can be understood using a simple pictorial approach. Interfacial electron transfer at electrode/solution interfaces: oxidation and reduction processes. Electron sink electrode (Anode). Electron source electrode (Cathode). P ne- Q Oxidation or de-electronation. P = reductant (electron donor) Q = Product A ne- B Reduction or electronation. A = oxidant (electron acceptor) B = Product • The greater the applied voltage, the larger the resulting current flow, and the greater the rate of the chemical reaction. • The rate at which charge is moved across the M/S interface = the rate at which chemistry is accomplished at the M/S interface. # electrons transferred Electrode area (cm2) Current (A) dq • In electrolysis we use an applied voltage i dt to perform chemistry at a M/S interface. • The applied voltage drives the chemical reaction which does not occur spontaneously. Charge (C) • The current flowing across the M/S interface is a measure of the rate of Time (s) the chemical transformation at the interface. nFA dN nFAf dt Amount of Faraday Material (mol) Constant (Cmol-1) Reaction flux (rate) mol cm-2s-1 Interfacial electron transfer processes are of two types : oxidation or de-electronation , and reduction or electronation. In oxidation, a species present in the electrolyte (termed a reductant or electron donor) may donate an electron to the electrode and in so doing become chemically transformed. The electrode acts as an electron sink and is termed an anode (fig. 1(a)). In reduction, an oxidant present in the electrolyte accepts an electron from the adjacent electrode, and in so doing becomes chemically transformed. In this case the electrode acts as an electron source and is termed a cathode (fig.1(b)) . Hence interfacial electron transfer has a chemical implication. Substances may be transformed and chemistry done at interfaces via application of an external electrical potential. We present some typical examples of oxidation and reduction reactions in the table across. We note from table 1 that redox reactions may be of many different types, and can vary from simple single step processes involving the transfer of a single electron, to complex multistep processes involving the transfer of many electrons. A further point should be noted here. Besides the elementary act of electron transfer itself, one must in many cases consider further complicating processes such as the occurence of chemical reactions preceding or following the electron transfer step (CE or EC processes), adsorption processes involving reactants, intermediates or products, and electrochemical nucleation and phase formation reactions. Furthermore the process of reactant transport to the electrode and product transport from the electrode will be important especially when the rate of electron transport is high. Fig. 1(a) Fig. 1(b) Electron sink electrode (Anode). Electron source electrode (Cathode). P ne- A ne- Q B Reduction or electronation. A = oxidant (electron acceptor) B = Product Oxidation or de-electronation. P = reductant (electron donor) Q = Product Table 1. Representative oxidation and reduction reactions. Cathodic reduction process Fe3 aq e Fe2 aq a 2H2 O 2e H2 g 2OH b O2 g 4H aq 4e 2H2 O c Cu 2 aq 2e Cus d 2CH2 CHCN 2H2 O 2e CH2 CH2 CN 2 2OH Anodic oxidation process Ce 3 aq Ce4 aq e a f Pt H2 O PtO 2H 2e 2Cl aq Cl2 g 2e g Fe s Fe 2 aq 2e h CH3OH H2 O CO2 6H 6e i e (a) Simple outer sphere single step single electron transfer reaction. This reaction type serves as a prototype in fundamental kinetic studies. (b) The cathodic evolution of hydrogen. A much studied multistep, multielectron transfer reaction which involves the formation of intermediate species adsorbed on the electrode surface. (c) The electrochemical reduction of dioxygen. Much studied in fuel cell technology. Again a complex reaction involving multiple electron transfer and the involvement of adsorbed intermediates. (d) Metal electrodeposition. An example of an electrochemical phase formation reaction involving nucleation and growth processes. (e) Conversion of acrylonitrile (AN) to adiponitrile (ADN). An example of electrochemistry applied to organic synthesis. (f) Metal oxidation. An example of a surface electrochemical redox transformation. Again this involves the generation of transient intermediates. (g) Chlorine gas evolution. Important in the chlor-alkali industry. This is the anodic analogue to hydrogen evolution. Again adsorbed intermediates are involved in the reaction mechanism. (h) Metal dissolution reaction. Fundamental basis of low temperature corrosion. (i) Oxidation of methanol. Much studied in fuel cell research. Typical example of a complex multistep , multielectron transfer process. In short, one may classify electrode processes into two general classes : simple single step charge transfer processes involving inorganic complex ions , and complex multistep multi-electron transfer reactions which consist of a number of elementary electron transfer and chemical reactions which can occur either in a consecutive or a parallel manner. Parallel electrode reactions are not common and are usually encountered in the anodic oxidation of certain organic compounds such as methanol in acidic or alkaline media. Consecutive processes are much more common. Typical examples of the latter type of processes are the cathodic oxygen reduction reaction, metal dissolution processes and various electro-organic transformations. Now analysis of the kinetics of simple single step reactions (i.e. outer sphere reactions involving inorganic complex ions which do not involve bond breaking) at a phenomenological level is relatively simple and is well understood. In contrast, the kinetics of multistep processes is generally complicated particularly for the case of electro-organic reactions. However some consecutive electrode reactions are particularly suitable for detailed attention since the number of steps involved is not too large. A particular example of the latter is the cathodic hydrogen evolution reaction. An important point to note from the table presented and from the previous discussion , is that many redox processes which underline important technological applications are complex. . Hence their study requires considerable ingenuity and the use of a broad arsenal of electrochemical and in situ spectroscopic techniques which are surface specific. Hence the modern approach to the study of electrode kinetics involves the use of electrochemical and non-electrochemical techniques. Indeed one may state that the examination of the kinetics and mechanisms of electrochemical reactions is one specific aspect of the broader area of surface science. The fundamental act in electrode kinetics, the interfacial electron transfer reaction, can be described in two ways. Firstly, one may adopt a macroscopic phenomenological approach, which results in the formal description of electron transfer kinetics in terms of rate equations and current / potential relationships. It is based largely on the activated complex theory of chemical reactions. Alternatively, one may consider a microscopic molecular based approach. This pathway leads to the subject of quantum electrode kinetics . Here an attempt is made to examine the molecular basis of interfacial electron transfer, and one seeks to describe the effect of the molecular structure of the reactant molecules and the electronic band structure of the electrode, on the kinetics of the electrode process. Quantum electrode kinetics was discovered early on in the development of electrochemistry, but was neglected for many years and consequently has not undergone the same degree of development as its more traditional macroscopic counterpart. At the present time there is little broad consensus in this topic. In this section we consider , in a very qualitative way, the kinetic description of a simple single step, single electron transfer process. A typical example might be the Fe2+(aq) / Fe3+(aq) redox reaction at a Pt electrode in an aqueous electrolyte solution . The latter type of reaction is termed an outer sphere electron transfer process since no bonds are broken or made during the course of the reaction. Consequently, we neglect complicating factors such as diffusional transport of reactants and products , and adsorption effects. How are the electron transfer kinetics examined experimentally ? It is obvious that one must do the measurement using an electrochemical cell containing at least two electrodes. Indeed it is conventional to use a three electrode arrangement as outlined across. The electrode of interest is termed the working or indicator electrode. The redox chemistry occurs at the interface between the working electrode and the electrolytic solution. A second electrode , termed the counter or auxiliary electrode, is required to complete the circuit . Current is passed in a circuit containing the working and the counter electrodes. Finally , one wishes to determine the potential difference across the working electrode / solution interface. The latter cannot be experimentally determined. What one can do however, is to measure changes in the potential of the working electrode with respect to a third electrode, the reference electrode, placed in the solution near the working electrode. The latter will have a very high impedance to current flow and so the potential of the reference electrode can be considered to be constant, irrespective of the current passed through the working and counter electrodes. Thus, the measured change in potential between the working and reference electrodes will be equal to the change in potential at the working electrode / solution interface. Hence in a typical experiment a potential waveform is applied to the working electrode , chemistry is done, and the resultant current response is monitored using a suitable output device . Current is passed between working and counter electrodes. The potential is measured between working and reference electrodes/. Reference electrode Working electrode Counter (auxillary) Electrode. The potential applied to the electrode is controlled using an electronic device called a potentiostat. It is now common practice to perform experiments under microcomputer control and to use the processing capabilities of modern microcomputers to collect, store and process the resultant data. How are the electron transfer kinetics examined experimentally ? It is obvious that one must do the measurement using an electrochemical cell containing at least two electrodes. Indeed it is conventional to use a three electrode arrangement .The electrode of interest is termed the working or indicator electrode. The redox chemistry occurs at the interface between the working electrode and the electrolytic solution. A second electrode , termed the counter or auxiliary electrode, is required to complete the circuit . Current is passed in a circuit containing the working and the counter electrodes. Finally , one wishes to determine the potential difference across the working electrode / solution interface. The latter cannot be experimentally determined. What one can do however, is to measure changes in the potential of the working electrode with respect to a third electrode, the reference electrode, placed in the solution near the working electrode. The latter will have a very high impedance to current flow and so the potential of the reference electrode can be considered to be constant, irrespective of the current passed through the working and counter electrodes. Thus, the measured change in potential between the working and reference electrodes will be equal to the change in potential at the working electrode / solution interface. Hence in a typical experiment a potential waveform is applied to the working electrode , chemistry is done, and the resultant current response is monitored using a suitable output device . The potential applied to the electrode is controlled using an electronic device called a potentiostat. It is now common practice to perform experiments under microcomputer control and to use the processing capabilities of modern microcomputers to collect, store and process the resultant data. A aq Baq e We consider the latter reaction which occurs at the interface between a metal and a solution. Let us assume that both A and B are present in the bulk of the solution with concentrations a∞ and b∞. We now consider the effect that charging the electrode (via application of an external potential) has on the rate of electron transfer. Let us firstly approach the situation at a qualitative level before we begin detailed kinetic analysis. Energy of electrons in metal increases upon application of a potential more negative than the thermodynamic equilibrium value. A net reduction (cathodic) current flows from metal to LUMO levels of redox active species in solution. n e- - LUMO LUMO EF HOMO Electron energy HOMO Redox couple in solution Metallic electrode Pictorial explanation of current flow due to reduction. Energy of electrons in metal decreases upon application of a potential more positive than the thermodynamic equilibrium value. Electron energy A net anodic (oxidation) current flows from the HOMO level of the redox species in solution to the metallic electrode. LUMO LUMO n e- EF HOMO HOMO + Redox couple in solution Metallic electrode Pictorial explanation of current flow due to oxidation. Consider the situation depicted in the previous two slides, one for a net reduction process and the other or a net oxidation process. In this pictures (which in effect distill some very complex quantum mechanical considerations) we indicate, in a very schematic way, the filled and empty electronic states in the metallic electrode and the highest occupied and lowest unoccupied energy levels of the donor species A and acceptor species B in the solution. The demarcation line between filled and empty electronic states in the metal is designated the Fermi energy EF. Now if species A and B are both present in solution and if no external potential is applied to the working electrode, then after a certain time a steady open circuit potential termed the equilibrium potential Ee may be measured. Indeed the value of this potential will depend on the logarithm of ratio of the concentrations of A and B via the Nernst equation as discussed previously in lecture 4. Under such conditions we may set EF = Ee. In contrast, when the working electrode becomes positively charged via application of an external potential more positive than Ee then the energy of the electrons in the metal will be lowered and EF shifts downwards in energy. If the applied potential is sufficiently positive then a stage will be reached such that EF is lower in energy than the HOMO level of the donor species A and one can obtain a net flow of electrons from the donor to the metal. An anodic oxidation current flows. Conversely, if a potential more negative than the equilibrium value is applied to the electrode then the energy of the electrons in the filled levels of the metal will be raised. A stage will be reached when EF is now higher in energy than the LUMO level of the acceptor species B and electrons will be transferred from the metal to B in solution. A cathodic reduction current will flow. We shall consider this picture in greater detail in the next chapter when we discuss the microscopic quantum mechanical approach to interfacial electron transfer. A survey of electrochemical reaction types. • Electrochemical reactions are usually complex multistep processes involving the • transfer of more than one electron. • In this course we focus on simple single step ET processes involving the transfer of a single electron. • The kinetics of simple ET processes can be understood using the activated complex theory of chemical kinetics (see SF Kinetics notes). Factors effecting the current/potential response at electrode/solution interfaces. • • • The current observed at an electrode/solution interface reflects two quantities: – Charging of electrical double layer : non Faradaic charging current iC – Interfacial ET across electrode/solution interface : Faradaic current iF. • Our aim is to be able to understand the shape that the general current/potential curve adopts and to be able to interpret the voltammogram in terms of ET and MT effects. We initially focus attention on the Faradaic component. The Faradaic current iF in turn can have components: at low potentials : arising from rate determining interfacial ET, – at high potentials : arising from material transport (MT) due to diffusion mechanisms. – These components can be quantified in terms of characteristic rate constants : k0 (units: cms-1) for ET and kD (units: cms-1) for MT. MT – ET Simple equivalent circuit representation of electrode/solution interface region. DL charging current iC CDL RS i Faradaic current iF Electrode Resistance of solution i iC iF RCT Solution Double layer charging current always presen in addition to Faradaic current in electrochemical measurements. Temperature effects in chemical kinetics. • Chemical reactions are activated processes : they require an energy input in order to occur. The same is true for electrochemical reactions which occur at electrode/solution interfaces. • Many chemical reactions are activated via thermal means. In contrast and in addition electron transfer reactions may also be activated via application of an electrode potential which alters the Fermi energy of the electrons in the electrode material. Activation • The relationship between rate constant k and temperature T is Energy given by the empirical Arrhenius equation (refer back to CH1101 Pre-exponential factor for details). The form of this equation is independent of the physical model used to explain it. The exact interpretation of the the pre-exponential factor A and the Activation energy Term EA • The activation energy EA is determined from experiment, by A measuring the rate constant k at a number of different temperatures. The Arrhenius equation is then used to construct an Arrhenius plot of ln k versus 1/T. The activation energy is determined from the slope of this plot. E k A exp RT The activation energy measures the sensitivity Of the rate constant to changes in temperature. d ln k d ln k RT 2 E A R d T dT 1 / ln k Slope 1 T EA R Van’t Hoff expression: Energy U 0 d ln K c 2 dT P RT TS E Standard change in internal energy: U 0 E E R P k E’ R U0 P k k Kc k d k d ln k d ln k U 0 dT ln k dT dT RT 2 P Reaction coordinate d ln k E dT RT 2 d ln k E dT RT 2 This leads to formal definition of Activation Energy. Arrhenius equation: more elaborate situations. In some circumstances the Arrhenius Plot is curved which implies that the Activation energy is a function of temperature. Hence the rate constant may be expected to vary with temperature according to the following expression. E k aT m exp RT We can relate the latter expression to the Arrhenius parameters A and EA as follows. ln k ln a m ln T E RT E d ln k 2 m E mRT E A RT 2 RT 2 dT T RT E E A mRT Hence E E k aT m e m exp A A exp A RT RT A aT m e m Svante August Arrhenius A more comprehensive Picture of chemical reactivity is given by Transition State Theory. Refer to earlier lectures Presented in this Module by Prof. Bridge. We begin by noting that chemical reactions are thermally activated processes. Consequently in order for chemical to occur, the reactant species must initially come together in a molecular encounter and then gain enough thermal energy to subsequently pass over the activation energy energy barrier . We recall that the activation energy is simply the energy required to bring the reactants to some critical configuration from which they can rearrange to form products. It is clear that during the course of a chemical reaction bonds will be stretched and broken in the reactants and new bonds will be formed. Hence we see that the potential energy of the system will vary during the course of the reaction. Hence a more complete and rigorous description involves the examination of potential energy changes during the course of a chemical reaction. This approach leads to the development of multidimensional potential energy surfaces where the potential energy of the system is plotted as a function of various bond distances and bond angles. Consider the following simple reaction between an atom A and a diatomic molecule BC in the gas phase : A BC ABC* AB C In the latter expression the reactants are transformed to products via formation of a high energy activated complex [ABC]* . If a potential energy surface is to be calculated for this reaction then the potential energy V should be plotted as a function of the two bond distances RAB and RBC and the bond angle = ABC . Of course, this procedure would necessitate use of a four dimensional diagram. However to make matters less complicated, one usually fixes one of the parameters, say , at a particular value, and then one examines the corresponding three dimensional surface defined by the axes V(R), RAB and RBC . The latter is a particular three dimensional cut from the four dimensional energy hypersurface. A particular three dimensional potential energy surface corresponding to a fixed value is presented in the next slide. We note that the course of the reaction is represented by a trajectory on the potential energy surface from P to Q, where P denotes the classical ground state of the diatomic molecule BC and Q represents the ground state of the diatomic molecule AB. Note that the variation of V(R) with bond distance R is described by the Morse potential energy function : V R Do 1 exp a R Re 2 where Do denotes the bond dissociation energy , Re is the equilibrium bond separation distance, and a is the anharmonicity constant. Activated complex Transition state Energy Now the system will tend to describe a trajectory of relatively low energy along the potential energy surface. Typically, this pathway involves two valleys meeting at a saddle point or col (labelled †) located in the interior of the potential energy surface. This pathway is outlined in more detail in the figure. Hence we see that in order for the system to pass from the reactant state to the product state, it will tend to travel along the bottom of the first valley (reactant region), over the col, and down into the second valley (product region). This minimum energy trajectory, termed the reaction coordinate , q, is illustrated as a dashed line in the figure presented across. This represents the most probable pathway along the three dimensional energy surface for the P / Q transformation. Finally, a cross section through this minimum energy path, known as a reaction profile is illustrated in the fig. outlined below. We pay particular attention to the point at the top of the V(q) profile. This point corresponds to the saddle point on the potential energy surface, and corresponds not only to a position of maximum energy with respect to the reaction coordinate, but also defines a position of minimum energy with respect to trajectories at right angles with respect to the reaction coordinate. Hence the activated complex or transition state theory focuses attention on the chemical species at the saddle point, i.e., at the point where the reactants are just about to transform into products. At this point one has the activated complex which is that special configuration of atoms of a system in transit between reactants and products. Ea Reaction coordinate Henry Eyring 1901-1981 Developed (in 1935) the Transition State Theory (TST) or Activated Complex Theory (ACT) of Chemical Kinetics. Reaction co-ordinate 0 Our presentation to date has been couched in Where G denotes the electrochemical Gibbs energy of qualitative terms. We can of course get more activation. Note that the term has got units of s-1 quantitative but this has already been covered and is expressed in cm. Hence k ' has got units of cm s-1 as it should. adequately earlier on in the module by Prof. Bridge. Now we have stated that chemical activation occurs via collisions between molecules and that reaction involves formation of a discrete activated complex of transient existence. The latter species represents a configuration in which the reactant molecules have been brought to a degree of closeness and distortion, such that a small perturbation due to a molecular vibration in an appropriate direction will transform the complex to products. The overall rate of reaction is then equal to the rate of passage of the activated complex through the transition state. The Activated Complex Theory has been discussed earlier on but we will deal with a special case of a unimolecular reaction at an electrode surface. Since the reaction is heterogeneous we must include a characteristic reaction layer thickness which we label (the length of which is of the order of a molecular diameter). Hence for an interfacial electron transfer reaction we have the following. kB T G 0 k exp h RT Phenomenological electrode kinetics In this section we consider the rate of electrochemical reactions. This is the domain of kinetics . Thermodynamics is very useful, it is a well developed subject, but it can only tell us whether a given reaction will occur in a spontaneous manner . In short thermodynamics sets the ground rules : it determines whether a given reaction is energetically feasible . If we wish to be able to determine how quickly an electrode reaction occurs we need to apply chemical reaction kinetics . The rate or velocity of an electrochemical reaction is expressed in electrical terms . Reaction rate is expressed as a current i . The quantity of charge passed across an interface can be directly related to the amount of material that has undergone chemical reaction as outlined via the following expression q nFAN k f k Ea0 (3) (1) In the latter expression q denotes the charge passed , n is the number of electrons transferred , A is the geometric surface area of the electrode, F is the Faraday constant and Nk represents the amount of species k chemically transformed per unit area of the electrode surface . hence the rate of reaction is given by : i This is a very important relationship . It expresses the connection between the rate at which charge is moved across the interface and the rate at which chemistry is done at the interface . The quantity fƩ denotes the reaction flux or reaction velocity of the heterogeneous electrochemical process , and is expressed in units of mol cm-2 s-1 . We recall that the velocity of a chemical reaction is proportional to the concentration of the reactants , and that the constant of proportionality is termed the rate constant . So it is also with heterogeneous electrochemical processes at interfaces . We consider a simple electrochemical transformation which occurs irreversibly at an ne electrode/solution interface A .The flux B or reaction velocity will be given by the following expression : dN dq nFA k nFAf dt dt (2) In the latter we note that a0 denotes the interfacial concentration of the reactant species A (units : mol cm-3 ) and k’E represents the heterogeneous electrochemical rate constant (units : cm s-1) . Note that the corresponding units for a first order chemical reaction occurring in the bulk of an aqueous solution are s-1 . The difference in units used in the electrochemical situation reflects the heterogeneous nature of the process . Charge is transferred between two phases of differing composition . Now we come to the crux of the matter . The heterogeneous electrochemical rate constant possesses a characteristic that is not exhibited by ordinary chemical rate constants . Its magnitude is strongly influenced by the magnitude of the potential applied to the electrode . Put another way : the rate of the electrode process is strongly influenced by the energy of the electrons near the Fermi level in the electrode material . By varying the energy of the electrons in the metallic electrode, one can significantly alter the rate of chemical transformation at the interface . This is a very significant result . We see that the electrode potential acts as a driving force for interfacial electron transfer . It is the objective of phenomenological electrode kinetics to derive a quantitative relationship between the heterogeneous rate constant and the driving force for electron transfer . Electrochemical reactions are thermally activated processes . In order for any chemical reaction to occur the reactant species must initially come together in a molecular encounter and then gain enough thermal energy to subsequently pass over the activation energy barrier . It is clear that during the course of a chemical reaction bonds will be stretched and broken in the reactants and new bonds will be formed when the products are generated . All that has been said so far applies to any chemical reaction . However for electron transfer processes at interfaces , the externally applied potential affects the interfacial potential difference or electric field at the interface , and this in turn has a marked effect on the activation energy barrier . It is the effect of the externally applied potential on the activation energy barrier which differentiates an electrochemical reaction from an ordinary chemical reaction . The rate of the latter can only be affected by temperature . Electrochemical reactions are not only temperature dependent, they are also potential dependent . The heterogeneous electrochemical rate constant depends on applied potential according in the following manner. F E E 0 0 k E k exp k exp RT 0 (4) In the latter expression k0 denotes the standard electrochemical rate constant which is a measure of the kinetic facility of an electrode process . If k0 is large than the ET kinetics will be fast and vice versa . The parameter is called the symmetry factor and determines how much of the input electrical energy fed into the system will affect the height of the activation energy barrier for the electrode process . Note that will exhibit values between 0 and 1 and in many cases will be close to 0.5 . Note that the positive sign in the exponential refers to an oxidation process and the negative sign to a reduction process . It is also possible to express the information contained in eqn.4 in the following way since kinetic expressions in electrochemistry may be expressed either in terms of rate constants or in terms of currents . Both representations are equivalent . F i i 0 exp (5) RT In the latter expression i0 is termed the exchange current density (note that current density is simply the current divided by the area of the electrode and so has units of A cm-2). This quantity is related to the standard heterogeneous rate constant k0 via : i 0 FAk 0 a 0 b 0 1 (6) Note also that denotes the overpotential . The latter quantity is defined as the extra potential , over and above that dictated by thermodynamics, which one must apply in order to make an electrode reaction occur at a net finite rate . Hence symbolically the overpotential is defined as : e E i E e (7) In the latter expression we recall that E(i) represents the potential measured with respect to a chosen reference electrode when a net current i is flowing and Ee is the potential recorded at equilibrium . We can see that the overpotential can be thought of as an example of the concept that one cannot get something for nothing . It is not possible for an electrode reaction to take place at a significant rate under conditions at which thermodynamics says it may occur . The deviation of actuality from the thermodynamic expectation is the overpotential . Overpotential has to be inputted into the system in the form of electrical energy in order to make the electrode reaction proceed at a finite rate . The larger the rate , the greater is the overpotential contribution necessary . Hence we note that: E(i) = - ref and Ee = e - ref , where ref denotes the Galvani potential of the chosen reference electrode . Let us now assume that the interfacial ET reaction is reversible . Hence we can write : k'E A B k'-E where we assume that the forward process is a deelectronation oxidation process and the reverse reaction is an electronation reduction process . Both processes involve the transfer of a single electron . The net flux is given by : f k E a0 k E b0 and the heterogeneous constants are given by : electrochemical k E k 0 exp k E k 0 exp 1 rate In the latter expressions we introduce the normalised potential as : F E E 0 0 RT Alternatively we can state F 1 F exp i i 0 exp RT RT This is the Butler-Volmer equation which is the fundamental equation of electrochemical ET kinetics . We recall that electrochemical equilibrium implied, on a microscopic level, that the electronation and deelectronation fluxes were in balance . No net currents passed across the electrode/electrolyte interface . Now if a net current is passed, this balanced situation is perturbed , a net current will flow in one specific direction . One has a departure from equilibrium . The greater the departure from equilibrium , the larger will be the observed net reaction rate and the larger will be the overpotential . Positive overpotentials correspond to oxidation processes , negative overpotentials to reduction processes . Zero overpotential corresponds to equilibrium . This can be summarised as follows. Net oxidation : > e 0 Net reduction : < e Equilibrium : = e 0 0 Transport and kinetics at electrode/solution interface • We consider two fundamental processes when considering dynamic events at electrode/solution interfaces: – Reactant /product transport to/from electrode surface – Electron transfer (ET) kinetics at electrode surface. • We first consider the kinetics of interfacial ET from a classical, macroscopic and phenomenological (non quantum) viewpoint. • This approach is based on classical Transition State Theory, and results in the Butler-Volmer Equation. DMT ETK • In electrochemistry the rate constant k varies with applied potential E because the Gibbs energy of activation G* varies with applied potential. Energy Amount of Barrier lowering F Transition state Activated complex Reaction Flux mol cm-2 s-1 =0 G0 * Reactant state G * Product state finite dN dq nFA nFAf i dt dt f k ' ET c0 Interfacial reactant Heterogeneous concentration ET rate constant mol cm-3 cm s-1 G * G0 * F Total added Electrical energy F Reaction coordinate Application of a finite overpotential lowers the activation energy barrier by a fixed fraction . Symmetry factor overpotential E EN Applied potential Thermodynamic Nernst potential We use the result of TST to obtain a value for the ET rate constant. f k ' ET c0 Apply Eyring eqn. from TST k ' ET G * Z exp RT k T Z B h Transmission coefficient overpotential G * G0 * F Electrochemical Gibbs energy of activation Characteristic ET distance (molecular diameter). Symmetry factor k 'ET G0 * F exp Z exp RT RT F 0 k ET exp RT The important result is that the rate constant for heterogeneous ET at the interface depends in a marked manner with applied electrode potential. As the potential is increased the larger will be the rate constant for ET. i nFAf Butler-Volmer Equation. For the moment we neglect the fact that mass transport may be rate limiting and focus attention on the act of electron transfer at the electrode/ solution interface. We examine the kinetics of a simple ET process in which bonds are not broken or made, involving the transfer of a single electron in a single step. Symmetry factor e A(aq ) B (aq ) e B(aq ) A(aq ) Normalised potential Net rate • Oxidation and i iox ired Reduction i i0 exp exp 1 processes are BV equation microscopically reversible. Exchange Reduction • Net current i at interface Oxidation • Exchange current current component reflects a balance between component provides a measure iox and ired . of kinetic facility Thermodynamic overpotential Nernst potential • Symmetry factor determines of ET process. how much of the input electrical energy fed into the system will 0 1 F F E E N Exchange i FAk a b 0 current affect the activation energy RT RT barrier for the redox process. Standard rate Note 0< < 1 and typically = 0.5. Applied constant potential Delineating the regions of the simple Butler-Volmer Equation. i i0 exp exp 1 reduction oxidation 15 i iox i0 exp Tafel Region 10 Linear Ohmic region 5 = i/i0 The Butler-Volmer equation describes the shape of the current density / overpotential characteristic of an interfacial ET reaction . Typical i versus curves are presented across for typical values of the exchange current density . Note that when = 0 the net current i density is zero and the reaction is at equilibrium . The current density is seen to rise rapidly when the overpotential deviates from its equilibrium value of zero . The rate of current increase with overpotential depends on the magnitude of the exchange current density and hence on the value of the heterogeneous standard rate constant . 0 i ired i0 exp 1 -5 -10 -15 -6 Tafel Region -4 -2 0 F/RT 2 4 6 1/ 2 i 2i0 sinh 2 The situation in terms of potential energy curves of the type used in transition state theory are presented across. ired iox i ired Net oxidation iox ired Net Reduction i iox 15 10 = i/i0 5 0 iox ired -5 i 0 -10 Equilibrium -15 -6 i iox ired -4 -2 0 F/RT 2 4 6 The shape of the current/potential curve depends on the numerical value of the symmetry factor. 100 0.9 80 60 0.7 = i/i0 40 0.5 20 0 -20 When differs from 0.5 the i vs curve becomes asymmetrical. -40 0.3 -60 0. 1 -80 -100 -6 -4 -2 0 F/RT 2 4 6 Approximations to the BV equation (I). High overpotential Tafel Limit. F 1 F The BV equation reduces to the exp i i0 exp Tafel equation when the overpotential RT RT is large, typically > 120 mV. iox ired At high overpotentials we assume that the ET reaction occurs in If >> 0 then i iox : net de-electronation the forward direction and the or oxidation. reaction occurring in the reverse F F direction can be neglected. i iox i0 exp ln i ln i0 This results in the derivation of a RT RT logarithmic relationship between If << 0 then i ired : net electronation or current and overpotential. reduction. A plot of ln i vs is linear. This is called a Tafel plot. 1 F i ired i0 exp Evaluation of the slope of the linear RT Tafel region enables the symmetry Factor to be evaluated, whereas the exchange Current i0 is obtained from the intercept at = 0. ln i ln i 1 F 0 RT Tafel Plot Analysis Linear Tafel Region 100 Oxidation log (i/i0)=log 10 log i Reduction 1 Linear Tafel Region 0.1 0.01 0.001 -8 d RT 2.303 bA F d log i d RT 2.303 bC 1 F d log i -6 -4 -2 0 = F/RT Exchange Current evaluated At = 0 2 4 6 8 Tafel approximation not valid at low overpotential Approximations to the BV equation (II). The low overpotential linear limit. F 1 F i i0 exp exp We note that the logarithmic RT RT Tafel behaviour breaks down as iox ired 0. Taylor expansion at small x. Tafel behaviour is characteristic of totally irreversible (hard driven) x2 expx 1 x ET kinetics and will only be valid 2! if the driving force for the x2 exp x 1 x electrode process is very large 2! which will be the case at high F F overpotentials. exp 1 RT RT In the limit of low overpotentials 1 F 1 F F 1 F exp 1 ( < 10 mV) the exponential terms RT RT RT RT in the BV equation may be simplified via use of a Taylor expansion to produce F F F i i 1 1 0 a linear relationship between current RT RT RT and overpotential (Ohm’s Law). F Linear approximation i i0 RT Linear i versus relation Charge Transfer Resistance 2 = i/i0 • We introduce the concept of a 1 charge transfer resistance RCT as being a measure of the resistance 0 to ET across the metal/solution interface. -1 • Since RCT and i0 are inversely proportional to each other, then a -2 large value of RCT implies a small -2 -1 value of i0 and vice versa. • Large RCT implies sluggish ET kinetics, iF and small RCT implies facile ET kinetics. i 0 • Useful idea : physical act of ET modelled RT in terms of a resistor of magnitude RCT. • Hence ET process described using an electrical equivalent circuit element. 0 1 2 = F/RT RT d RCT di i 0 i0 F iRCT Ohms Law RCT Interfacial ET Transport effects in electrode kinetics • Influence of reactant transport (logistics) becomes important when the applied overpotential becomes very large. • The current/potential curve “bends over” and a current plateau region is observed. • This observation is explained in terms of rate control via diffusion of the reactant species to the electrode surface. • We consider a two step sequence : diffusion to the site of ET at the electrode/solution interface followed by the act of ET itself. • When the overpotential is very large, the driving force for interfacial ET is very large, and so ET becomes facile and hence no longer controls the rate. • Matter transport via diffusion (i.e. getting the reactant species to the region of reaction) becomes rate limiting. Transport and kinetics at electrode/solution interfaces Low overpotential situation: Charge Transfer Control ETK DMT Large overpotential situation : Mass Transport Control DMT ETK Bulk Phase Reactant Bulk Phase Reactant Reactant at electrode Fast diffusive mass transport, slow rate determining interfacial ET kinetics Product at electrode Reactant at electrode Slow rate determining Diffusive mass transport, Fast interfacial ET kinetics Product at electrode Transport and kinetics at electrodes. Current density 1.0 i J nFA nF Net flux ET & MT D kD = i / iD f Diffusion layer approximation used. 0.5 MT D dc f D k ET c0 c c0 k D c c0 dx 0 0.0 -6 -4 -2 0 2 4 6 = F(E-E0)/RT ET c c0 k 1 ET D k ET k D c f k D k ET f k ET c0 k ET k exp 0 Normalised potential 1 1 1 f k ET c k D c Mass transport corrected Tafel Equation k ET k D c kD c fD i f nFA k ET k D 1 k D 1 k D k ET k ET F E E k ET k 0 exp RT fD iD kDc nFA fD k 1 D f k ET 0 MT corrected Tafel plot i i ln D i F E E0 iD fD f D f kD 1 1 0 exp i f f k RT f D f iD i kD F E E ln ln ln 0 f i RT k 0 k ln( D0 ) k ln k D 0 k S F RT E E0 This is one form of the mass transport corrected Tafel equation. We see that a plot of ln (iD-i/i) vs – is linear . The slope of this plot yields - F/RT whereas the intercept directly yields ln (kD/k0). Since kD may be readily evaluated, then the standard rate constant k0 may be determined. This form of Tafel plot has been much used in the literature. Other useful data analysis strategies. The Tafel analysis may be applied with good accuracy to obtain i0 and when the exchange current density for the electrode process is low (typically when i0 < 10-3 A cm-2). In contrast , the low overpotential linear approximation is useful when the exchange current density is large (i0 > 10-3 A cm-2). i0 FAk 0 a 1 b A ne B ln i0 ln FAk 0 1 ln a ln b ln i0 1 ln a b ln i0 lnb a Hence we see that a plot of ln i0 versus either ln a∞ or ln b∞ should be linear with slope producing a value for the symmetry factor . We note that the Butler-Volmer equation may be used to determine i0 and regardless of the magnitude of the overpotential using the following procedure. i F F F F ln ln i i i0 exp exp exp 0 F RT RT RT RT 1 exp RT F F i0 exp 1 exp RT RT Allen-Hickling Equation Hence we see that a plot of ln i 1 exp F RT versus should be linear, the slope yielding and the intercept at = 0 yielding i0 . The molecular interpretation of electron transfer. We have presented a brief analysis of the fundamentals of electrode kinetics in terms of current/overpotential relationships . The analysis was based on macroscopic or phenomenological considerations . However a proper understanding of interfacial electron transfer requires the adoption of a microscopic perspective . This requires quantum mechanics . The sharp rise in current density with increasing overpotential as presented in the current vs overpotential curve can be understood as follows . Here we present a molecular interpretation of interfacial electron transfer . It is a quantum mechanical process and a quantitative development requires some sophisticated mathematics . Consequently we shall adopt the lazy mans approach and present a qualitative pictorial presentation of the essentials . Consider the situation depicted in the next two slides which we have shown previously. In these pictures we indicate , in a very schematic manner, the filled and empty electronic states in the metallic electrode and the highest occupied and lowest unoccupied energy levels of the donor species A and acceptor species B in the solution . The demarcation line between filled and empty electronic states in the metal is designated the Fermi energy EF . Now if the redox species A and B are both present in the solution and if no external potential is applied to the metal, then as previously noted, an equilibrium potential Dfe or Ee will be set up reflecting the balanced Faradaic activity across the interface . The value of the latter potential will depend on the logarithm of the ratio of the activities of A and B via the Nernst equation as previously discussed . Under such conditions we may set EF = Ee . In contrast, when the electrode becomes positively charged via application of an external potential more positive than Ee , then the energy of the electrons in the metal is lowered and EF shifts downward in energy . If the applied potential is sufficiently positive then a stage will be reached such that EF is lower in energy than the HOMO level of the donor species A and one obtains a net flow of electrons from donor species to metal . An anodic oxidation current flows . Conversely, if a potential more negative than the equilibrium value is applied to the electrode, then the energy of the electrons in the filled levels of the metal will be raised . A stage will be reached when EF is now higher in energy than the LUMO level of the acceptor species B and electrons will be transferred from the metal to species B in solution . A cathodic reduction current will flow . Energy of electrons in metal decreases upon application of a potential more positive than the thermodynamic equilibrium value. Electron energy A net anodic (oxidation) current flows from the HOMO level of the redox species in solution to the metallic electrode. LUMO LUMO n e- EF HOMO HOMO + Redox couple in solution Metallic electrode Pictorial explanation of current flow due to oxidation. Energy of electrons in metal increases upon application of a potential more negative than the thermodynamic equilibrium value. A net reduction (cathodic) current flows from metal to LUMO levels of redox active species in solution. n e- - LUMO LUMO EF HOMO Electron energy HOMO Redox couple in solution Metallic electrode Pictorial explanation of current flow due to reduction. The discussion has focused on the reactant species , the electron donor or electron acceptor . Bonds become stretched or activated , the potential energy of the system changes and, at a critical configuration (the transition state) the electron is somehow magically transferred . The following question arises : how does the electron reach the acceptor species in solution , or conversely, how does the electron pass from the donor to the metal ? The answer proposed from classical physics is that the electron would be emitted from the metal, passing directly over the potential energy barrier at the metal/solution interface . However it can be shown that an electron emission of this type would only generate a very tiny current , far less than that observed experimentally . The problem is not unlike the situation of a particle decay : radioactive substances do decay , a particles leave the nucleus although classically they are not meant to do so . What does happen is that the electron (and indeed the a particle) does not pass over the potential energy barrier, but actually tunnels through the barrier from their levels in the metal to the vacant levels of the acceptor ion in solution . Hence we state : the fundamental act of interfacial electron transfer is a quantum event, governed by the rules of quantum mechanics . It is possible (Bockris and Khan 1983) to quantitatively evaluate the tunnelling probability PT by solving from first principles or approximately (via the WKB approximation ) the pertinent Schrodinger equation provided that a suitable barrier geometry is proposed . Typical barrier shapes used in calculations of this type are rectangular or parabolic or those of the Eckart type . For instance if we assume that the electron tunnels through a rectangular potential energy barrier and described by a potential energy function of the form : U( x) 0 x 0 U( x) U 0 0xa U( x) 0 x a In the latter a represents the width of the rectangular barrier and U0 denotes the barrier height , Total energy of particle Hence we see that the overpotential for the production of a definite reaction rate at a metal/solution interface is in fact the shift of the Fermi Level in the metal from the value which it had when the electrode reaction was taking place at equilibrium . Incidence of particle on barrier Transmission of particle Through barrier Barrier width a I III II Barrier Height U0 Distance 0 a Now if the energy E of the transferring electron is less than U0 , detailed solution of the pertinent Schrodinger equation results in the following expression for the tunnelling probability which we label PT : sinh 2 ka PT 1 4 E E 1 U0 U 0 1 k In particular when the barrier is very high (U0 / E >> 1) and wide (so that ka >> 1) we can write that 2 provides a measure of the ‘opacity’ of the barrier , and in the classical limit PT will be very small due to the fact that the opacity parameter will be very large . Barrier Type Barrier Heigth (wrt Fermi Level /eV Barrier Width/Å Tunneling Probability Parabolic Barrier 5 5 1.2 x10-4 10 10 9.1 x10-12 15 15 5.2 x10-21 20 20 6.0 x10-32 5 5 1.0x10-5 10 10 8.8x10-15 15 15 1.5x10-26 20 20 1.7x10-40 and so the tunnelling probability PT reduces to : Rectangular Barrier 16E 16E E exp 2 ka exp 2 ka 1 U0 U0 U0 For a parabolic barrier one can show that The tunneling probability takes the following form Where L = barrier width and U0 is barrier maximum. 2L 1/2 PT exp 2m U 0 E h 2m U 0a 2 2 2 m U 0 E sinh ka exp ka 2 PT The latter expressions lead to the remarkable prediction that a quantum particle exhibits a certain finite (although small) probability of leaking through a potential energy barrier which is completely opaque from the viewpoint of classical physics (since E < U0) . Indeed the parameter We note that the tunneling probability is very sensitive to the barrier parameter. A further consideration must be noted . In order for electron tunnelling to occur, the electron must move to an acceptor energy level exactly equal in energy to that of the electron in the metal . In other words the electron transfer process is said to be radiationless . These acceptor states are of a special kind : they are said to be vibrationally excited . We can quote a concrete example at this point . In the electronation of a hydrated proton, which is one of the constituent steps in the multistep hydrogen evolution reaction, The latter reaction is often used as a prototype in the fundamental investigation of electrode processes . + H 3O + e + M MH + H2O In the latter expression M denotes a vacant metal site and MH an adsorbed hydrogen atom , one supposes that the O-H bonds which receive electrons are in excited vibrational states . The degree to which this excitation energy exceeds the ground state energy of the reaction is a contribution to the activation energy . Hence molecular activation via bond stretching is a very important contribution to the net activation energy . One can also show that the solvation of the ion also plays an important role . The ion in solution has associated with it a solvation shell . The solvation state of the reduced and oxidised form of a redox couple is different . Hence the solvation shell of the reactant species must be configured in a certain way to allow the electron transfer reaction to occur . This re-organization of the solvation shell also requires energy and so will also be a contributor to the energy of activation . These ideas form the basis for the Marcus theory of interfacial electron transfer. The Marcus model assumes that the electron transfer rate constant is given by : G k E A exp k BT where G* is the Gibbs energy of activation . The preexponential factor is given by A e n n (Sutin 1982 ; Hupp and Weaver 1984 ) where e is the electronic transmission coefficient and represents the probability with which electron transfer will occur once the transition state has been formed ( and so is equivalent to the tunnelling probability) . Typically 0 e 1 . Also n (units : s-1 ) is the nuclear frequency factor and is defined as the frequency at which the configuration of nuclear co-ordinates appropriate for electron transfer is attained . In other words it represents the rate at which the reactant species in the vicinity of the transition state is transformed into products . This factor depends on the mechanism of activation and will include contributions from solvent re-organization and bond stretching . Its value is typically in the range 0.5 - 1.0 x 1013 s-1 . Finally n is termed the nuclear tunnelling factor and is a quantum mechanical correction which becomes important at low temperature . All of these factors have been discussed in detail in a recent review (Weaver 1987) . 2e e k E A exp A exp exp 4 k B T 4k BT 2k BT Detailed development of this Marcus model which is based on the transition state theory of chemical reactions results in the following expressions for the heterogeneous electron transfer rate constants : e 2 k E A exp k T 4 B k E e 2 A exp 4k B T where A is a pre-exponential factor the nature of which has been discussed by Hupp and Weaver (1983) and denotes the re-organization energy which consists of both an outer sphere and an inner sphere component reflecting the two contributions (solvent shell re-organization and bond stretching) to the activation energy . The latter quantity can be evaluated approximately and is typically 0.5 to 1.5 eV We note that if e the quadratic term in the energy of activation presented in the Marcus-Hush expressions outlined above may be expanded to first order in e and the heterogeneous electron transfer rate constant for the forward de-electronation process admits the form given on the left hand side across. This expression is just the simple Butler-Volmer equation with the symmetry factor having a numerical value of ½ . More generally according to the Marcus formulation the symmetry factor or transfer coefficient is given by : G 1 G 0 1 2 G 0 2 where we note that G* is the activation energy for electron transfer and G0 represents the standard Gibbs energy change on proceeding from the reactant to the product state which for an electrochemical reaction is given by - e . The Gibbs energy of activation is given by (Albery 1975) : G 0 G 0 * G 1 4 4 2 2 The expression for the symmetry factor presented in the expression above contains some important information . In general the latter quantity is a measure of the location of the transition state along the reaction co-ordinate . As outlined in figure 2.23 we can consider a number of simple situations . If, for example the transition state is symmetric then G0 = 0 and = ½ . Alternatively when G0 is negative the electron transfer proceeds in an energetically downhill manner and < ½ and the transition state is “reactant like” . Finally when G0 is positive , the electron transfer reaction is uphill and > ½ and the transition state is “product like” . The latter statements are a quantitative way of expressing the so called Hammond Postulate. Note also that the facility of electron transfer will depend on the magnitude of the re-organization energy . When G 0 e the electron transfer kinetics are slow and as noted in eqn.2.52. On the other hand when l is small and one has fast electron transfer then the following three cases must be considered . When then . Alternatively, if G 0 then 1 . Finally, if G 0 then 0 .
