# 快速时时彩: Reaction mechanism and rate determining steps.pdf

Reaction mechanism and rate determining steps S. Trasatti University of Milan, Milan, Italy 1 KINETIC ASPECTS Let us consider an electrochemical cell with both electrodes A and B in electrochemical equilibrium. The potential dif- ference between the two terminals, Delta1E = E B ? E A , mea- sures the difference in electronic energy between the two metals of the electrodes. Delta1E is determined by the nature of the electrochemical reactions taking place at the electrode interfaces. E B and E A are measured with respect to the same reference electrode, i.e., on the same potential scale, e.g., standard hydrogen electrode (SHE), saturated calomel electrode (SCE), reversible hydrogen electrode (RHE), etc. According to current conventions, the electrode with the more positive E, e.g., electrode B, is “the positive pole” and the other, A, is the “negative pole”. This terminology is applied to power sources, i.e., to electrochemical cells where only spontaneous transformations take place. The electronic energy is more negative in the positive pole than in the negative pole. If the terminals are connected through an external circuit (a simple variable resistor), electrons spontaneously flow from electrode A to electrode B, as in the case of a liquid going from a filled to an empty reservoir. The driving force Delta1E is sustained by the chemical potential energy stored in the cell as chemicals: Delta1E =? Delta1G nF (1) Equation (1) is the fundamental equation of electrochemical energy conversion. The flow of electrons in the external circuit promotes the departure from equilibrium of the electrode reactions whose proceeding will maintain Delta1E constant as long as reactants are available. This qualitative picture of the way electrochemical cells operate describes the real situation, but it does not highlight kinetic aspects. In fact, Delta1E is maintained only if the current in the external circuit is vanishingly small, i.e., the electrode reactions proceed reversibly. In reality, electrode reactions are activated as any other chemical reactions. Note that the simple act of electron transfer between two species is not activated, i.e., it proceeds radiationless. It is the fluctuations in the molecular configuration of reactants and products that determine the activation energy. Activation energies consume part of the driving force Delta1E; it follows that the reaction rate, i.e., the flow of elec- trons in the external circuit, can be increased proportionally with the departure of the potential difference between the terminals from the equilibrium value. The difference in electrode potential between the value at a given current density, j, and the value at equilibrium is what is known as the overpotential: η = E j ? E eq (2) Upon departure from equilibrium, the value of the posi- tive electrode decreases and that of the negative electrode increases: Delta1V = Delta1E ? Sigma1η (3) where Sigma1η is the sum of the overpotentials for the two electrodes. Since the electric power generated by an elec- trochemical cell is Delta1V × I, it is evident from equa- tion (3) why power sources with high Delta1E and small η are sought after. Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley Delta1V Omega1 is essentially a problem of cell design (engineering), but it may be related to the nature of the electrolyte and in some cases of electrode materials; Delta1V t is fundamentally related to materials chemistry and physics. 2 THEORY OF OVERPOTENTIALS Overpotentials in equation (5), Delta1η, account for energy losses related to activation energies and concentration pro- files. Two main kinds of overpotential are thus defined: η act (activation or barrier overpotential) and η conc (concentration overpotential). 2.1 Activation (barrier) overpotential The classical theory of η act is based on the model of activated complex with quasi-parabolic energy curves of reactants and products that move vertically without chang- ing shape as the energy of particles is changed by modifying the electrode potential. This is tantamount to assuming a linear variation of the electric field with the reaction coor- dinate. Furthermore, if the entropy and the heat capacity of the systems are assumed to be unaffected by the shifts in the curves, Gibbs energy curves reflect directly the behavior of energy curves. [1] For a single step electrode reaction involving n electrons, the change in energy brought about by an overpotential, η,is?nFη. The model leads to a linear variation of the activation energy with η conventionally given by αnFη and (1 ? α)nFη for the cathodic and the anodic reactions, respectively. For a process in a single step, α turns out to be the fraction of electrical energy transferred to the cathode activation energy. For this reason α is also called the “transfer coefficient”. α is defined as the “symmetry factor” because its value is determined by the ratio of the slopes of the energy curves at the intersection point. As a further meaning, α can also be described as the fraction of charge transferred to the reactant at the position of the activated complex. It is identified as a partial charge transfer coefficient. It is evident that in single step reactions transfer coefficient and symmetry factor coincide. Always in terms of classical theory of reaction rates, the current (density) is exponentially related to the activation energy: j = const k i exp parenleftbigg ? αnFη RT parenrightbigg (6) where k i is the “intrinsic” reaction constant (at equilibrium) and the equation has been written for an hypothetic cathodic reaction. For an anodic reaction: j = const k i exp bracketleftbigg (1 ? α)nFη RT bracketrightbigg (7) Often, for the sake of uniformity in formulations, the convention is used α c ≡ α and α a ≡ (1 ? α) with α a + 2 Reaction mechanism and rate determining steps α c = 1 so that equations (6) and (7) differ only for the sign of the exponential term. Again by convention, the current associated with oxida- tion reactions is taken as positive. Thus, the net current (density) for a reversible electrode reaction is j = j an ? j cat (8) and is positive as |j an | |j cat |. Since at equilibrium the partial anodic current equals the partial cathodic current j an = j cat = j 0 (9) where j 0 is the “exchange current”. From equations (6)–(9) the classical Butler–Volmer (BV) equation ensues: j = j 0 bracketleftbigg exp parenleftbigg (1 ? α)nFη RT parenrightbigg ? exp parenleftbigg ? αnFη RT parenrightbiggbracketrightbigg (10) which is at the base of the treatment of electrochemical kinetics. Near equilibrium, equation (10) reduces to a linear dep- endence of j on η which is called “reaction (charge transfer) resistance” since it formally reproduces Ohm’s law: j = j 0 nFη RT (11) Within |η|≤5 mV, such linear dependence can in principle offer a way to derive j 0 provided n is known and the reaction takes place in a single step. At higher overpotentials (|η|≥50 mV), depending on the sign of η, one of the terms in equation (10) becomes negligible with respect to the other and the equation reduces to equation (6) or (7), respectively. In terms of exchange current j a = j 0 exp parenleftbigg (1 ? α)nFη RT parenrightbigg , j c = j 0 exp parenleftbigg ? αnFη RT parenrightbigg (12) which are universally known, in semilogarithmic form, as Tafel lines: η = RT (1 ? α)nF ln j ? RT (1 ? α)nF ln j 0 (13) η = RT αnF ln j 0 ? RT αnF ln j(14) for anodic and cathodic reactions, respectively. Tafel lines possess a powerful diagnostic character for reaction mechanisms since the value of the slope (b = |RT/αnF|, always assuming a single step reaction) acquires well defined values for well defined mechanisms. Besides, Tafel lines allow the linear section to be extrapolated to η = 0, i.e., to E eq , with the intercept giving the exchange current, j 0 . This procedure assumes that the reaction mech- anism does not change within the potential range of extra- polation. 2.2 Activationless and barrierless reactions While the most probable value for α is 0.5 (symmetric energy barrier or electrical energy equally bearing on the anodic and the cathodic activation energy), other values are also possible depending on the details of the electrode reac- tion. The two limiting values α = 0andα = 1 have been discussed by Krishtalik. [2] If α = 0, it implies that a vari- ation in electrode potential does not produce any variation in the (cathodic) activation energy. This means that there is no more activation energy (activationless reaction). Con- versely, α = 1 implies that a variation in electrode potential bears entirely and exclusively on the (cathodic) activation energy. This implies that there exists no barrier which is responsible for the splitting of the applied overpotential between anodic and cathodic partial reaction (barrierless reaction). It is obvious that if the forward reaction is acti- vationless, the backward reaction is barrierless, and vice versa (Figure 1). For α = 0andα = 1, there are two extreme situations that can occur only in a restricted range of overpotential, in most of the cases outside the experimentally observ- able potential window. For this reason, evidence for such situations is usually weak requiring very accurate experi- mental data. The value of α = 1 can occur as the reactant state is at a much lower energy than the product. Thus, it can be observed only at low overpotential and very low current densities. Conversely, α = 0 can occur as overpotential is increased to such an extent that the whole activation energy has been balanced. Thus, the energy state of the reactant is at a much higher level than that of the product. In both cases a large difference in energy states between reactant and product is typical of these situations. 2.3 Concentration overpotential While η act cannot be avoided because it is intrinsic to the electrode reaction, the other (most common) form of overpotential, η conc , can be minimized with suitable expedients. The concentration overpotential is related to the concentration profiles which are formed at an electrode interface as the given species is consumed (or produced) at 3 Theory of electrocatalysis P otential energy Barrierless α=1 Distance Normal α=0.5 α=0 Activationless Figure 1. Sketch of the potential energy curves for reactants and products (the direction of the reaction is indicated by an arrow) with the system in normal (α = 0.5) and extreme (α = 0; 1) configurations. Drawn after Krishtalik (1968). [2] the electrode surface. The concentration gradient produces a diffusion flux towards (or away from) the electrode, which sustains the electrode reaction. In the former (most common) case: j = nFD i (c i ? c ? i ) δ (15) where i is the reacting species, c ? i is its concentration at the reacting site, D i its diffusion coefficient and δ the thickness of the diffusion layer. As c ? i → 0 j → j lim . η conc is not related to a physical energy barrier, thus, the physical meaning and implications are quite different from η act . η conc ensues from the computation of the energy loss with respect to the ideal situation (c ? i = c i ). In other (qualitative) words, at constant overpotential the reaction rate is lower in the presence of concentration gradients because the actual reactant concentration is lower than it could ideally be. The energy balance is based on simple thermodynamic considerations: η conc = RT nF ln parenleftbigg c ? i c i parenrightbigg (16) Since in equation (15) j lim ∝ c i and j ∝ c i ? c ? i , equa- tion (16) is transformed into η conc = RT nF ln parenleftbigg j lim ? j j lim parenrightbigg (17) which allows us to calculate η conc once j lim is known. If η act and η conc are both operative, η conc can be corrected using equation (17) and the Tafel line can be singled out from the experimental data. 2.4 Pseudo-overpotentials 2.4.1 Double layer effects While dynamic concentration profiles are responsible for the onset of η conc , static concentration profiles of charged species in solution can arise in the double layer region as a consequence of coulombic forces emanating from the charged electrode surface. These concentration profiles influence the kinetics of the electrode reaction in that both the local reactant concentration (c ? i ) and the electric potential at the reacting site (φ ? ) differ from the ideal ones (c i and φ S , respectively). These effects do not bring about a new form of overpotential, but more simply they distort the relationship between η and j, i.e., the Tafel line. [3] Such a picture is also known as the “Frumkin effect”. [4] Double layer effects can be corrected if double layer parameters are known (e.g., potential at the outer Helmholtz plane, which is regarded as the reacting plane of nonspecif- ically adsorbed ions). Thus, c i should be replaced by c ? i , related to φ by the Gouy–Chapman theory: c ? i = c i exp parenleftbigg ? ziFφ RT parenrightbigg (18) and the electrode potential, E, should be corrected for the part of potential which is not operating on the reacting site, i.e., (φ ? ? φ S ), reduced for simplicity to φ ? by set- ting φ S = 0. Corrections of the kinetic equation for a reacting particle of charge number z exchanging n electrons result in the 4 Reaction mechanism and rate determining steps following equation: j obs = j id exp parenleftbigg (αn ? z)Fφ ? RT parenrightbigg (19) which allows from observed currents to obtain back the cur- rent purged from double layer effects. Convincing examples of this situation are available in the literature. [5] Double layer effects on nonspecifically adsorbed ions are the simplest case. The situation becomes more and more complex in the presence of: (a) specifically adsorbed ions of the supporting electrolyte; (b) induced adsorption of reacting ions on specifically adsorbed electrolyte ions; (c) specific adsorption of reacting ions. In all cases a distortion of the kinetic data occurs but correction of double layer effects is hard because of the lack of a specific theory. 2.4.2 Ohmic drop effects Tafel lines are distorted if ohmic drops are incorporated into the measured electrode potential (cf. equation (5)) with the addition of a pseudo-overpotential term. Ohmic drops are minimized but not eliminated with the use of a Luggin capillary. If precise potential measurements are needed at high currents, the correction of ohmic drop effects is called for. This can be accomplished either instrumentally or graphically. The most adequate way is by means of the current interruption method at each value of the current. If the instrumentation does not allow for that, a graphic correction is possible more with (a) or less empirically with (b). (a) If in the plot of E vs. ln j a linear section can be identified in the low current range, this can be extended into the high current region and used as the actual Tafel line. The reliability of the approach can be checked by deter