A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 26
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Specifically, v = i/nFA. We also know that for a given electrodeprocess, current does not flow in some potential regions, yet it flows to a variable degreein others. The reaction rate is a strong function of potential; thus, we require potentialdependent rate constants for an accurate description of interfacial charge-transfer dynamics.In this chapter, our goal is to devise a theory that can quantitatively rationalize theobserved behavior of electrode kinetics with respect to potential and concentration. Onceconstructed, the theory will serve often as an aid for understanding kinetic effects in newsituations.
We begin with a brief review of certain aspects of homogeneous kinetics, because they provide both a familiar starting ground and a basis for the construction,through analogy, of an electrochemical kinetic theory.3.1 REVIEW OF HOMOGENEOUS KINETICS3.1.1Dynamic EquilibriumConsider two substances, A and B, that are linked by simple unimolecular elementary reactions.1A^B(3.1.1)4Both elementary reactions are active at all times, and the rate of the forward process,Vf (M/s), is(3.1.2)whereas the rate of the reverse reaction is"b = hCB(3.1.3)The rate constants, kf and k\>, have dimensions of s~\ and one can easily show that theyare the reciprocals of the mean lifetimes of A and B, respectively (Problem 3.8).
The netconversion rate of A to В isvnet = kfCA-kbCB(3.1.4)*An elementary reaction describes an actual, discrete chemical event. Many chemical reactions, as written, arenot elementary, because the transformation of products to reactants involves several distinct steps. These stepsare the elementary reactions that comprise the mechanism for the overall process.Я788 • Chapter 3. Kinetics of Electrode ReactionsAt equilibrium, the net conversion rate is zero; hence£ = * = §*^b(3.1.5)CAThe kinetic theory therefore predicts a constant concentration ratio at equilibrium, just asthermodynamics does.Such agreement is required of any kinetic theory.
In the limit of equilibrium, the kinetic equations must collapse to relations of the thermodynamic form; otherwise the kinetic picture cannot be accurate. Kinetics describe the evolution of mass flow throughoutthe system, including both the approach to equilibrium and the dynamic maintenance ofthat state. Thermodynamics describe only equilibrium.
Understanding of a system is noteven at a crude level unless the kinetic view and the thermodynamic one agree on theproperties of the equilibrium state.On the other hand, thermodynamics provide no information about the mechanismrequired to maintain equilibrium, whereas kinetics can be used to describe the intricatebalance quantitatively. In the example above, equilibrium features nonzero rates of conversion of A to В (and vice versa), but those rates are equal. Sometimes they are calledthe exchange velocity of the reaction, u 0 :We will see below that the idea of exchange velocity plays an important role in treatmentsof electrode kinetics.3.1.2The Arrhenius Equation and Potential Energy Surfaces (1,2)It is an experimental fact that most rate constants of solution-phase reactions vary withtemperature in a common fashion: nearly always, In к is linear with 1/Г.
Arrhenius wasfirst to recognize the generality of this behavior, and he proposed that rate constants beexpressed in the form:(3.1.7)where £ д has units of energy. Since the exponential factor is reminiscent of the probability of using thermal energy to surmount an energy barrier of height E&, that parameter hasbeen known as the activation energy. If the exponential expresses the probability of surmounting the barrier, then A must be related to the frequency of attempts on it; thus A isknown generally as the frequency factor. As usual, these ideas turn out to be oversimplifications, but they carry an essence of truth and are useful for casting a mental image of theways in which reactions proceed.The idea of activation energy has led to pictures of reaction paths in terms of potential energy along a reaction coordinate.
An example is shown in Figure 3.1.1. In a simpleunimolecular process, such as, the cis-trans isomerization of stilbene, the reaction coordinate might be an easily recognized molecular parameter, such as the twist angle about thecentral double bond in stilbene. In general, the reaction coordinate expresses progressalong a favored path on the multidimensional surface describing potential energy as afunction of all independent position coordinates in the system.
One zone of this surfacecorresponds to the configuration we call "reactant," and another corresponds to the structure of the "product." Both must occupy minima on the energy surface, because they arethe only arrangements possessing a significant lifetime. Even though other configurationsare possible, they must lie at higher energies and lack the energy minimum required for3.1 Review of Homogeneous Kinetics89ReactantsProductsFigure 3.1.1 Simple representationof potential energy changes during aReaction coordinatereaction.stability. As the reaction takes place, the coordinates are changed from those of the reactant to those of the product.
Since the path along the reaction coordinate connects twominima, it must rise, pass over a maximum, then fall into the product zone. Very often,the height of the maximum above a valley is identified with the activation energy, either£ A f or EA>b, for the forward or backward reaction, respectively.In another notation, we can understand E& as the change in standard internal energyin going from one of the minima to the maximum, which is called the transition state oractivated complex. We might designate it as the standard internal energy of activation,AE*. The standard enthalpy of activation, A//*, would then be Д£* + A(PV)*, but A(PV)is usually negligible in a condensed-phase reaction, so that ДЯ* « Д£*.
Thus, the Arrhenius equation could be recast as(318)We are free also to factor the coefficient A into the product А' cxp(AS*/R), becausethe exponential involving the standard entropy of activation, Д£*, is a dimensionless constant. Then( 3 1 9 )orwhere AG* is the standard free energy of activation.2 This relation, like (3.1.8), is reallyan equivalent statement of the Arrhenius equation, (3.1.7), which itself is an empiricalgeneralization of reality. Equations 3.1.8 and 3.1.10 are derived from (3.1.7), but only bythe interpretation we apply to the phenomenological constant E&. Nothing we have written so far depends on a specific theory of kinetics.2We are using standard thermodynamic quantities here, because the free energy and the entropy of a species areconcentration-dependent.
The rate constant is not concentration-dependent in dilute systems; thus the argumentthat leads to (3.1.10) needs to be developed in the context of a standard state of concentration. The choice ofstandard state is not critical to the discussion. It simply affects the way in which constants are apportioned inrate expressions. To simplify notation, we omit the superscript "0" from A£*, Д//*, AS*, and AGT, butunderstand them throughout this book to be referred to the standard state of concentration.90Chapter 3. Kinetics of Electrode Reactions3.1.3Transition State Theory (1-4)Many theories of kinetics have been constructed to illuminate the factors controlling reaction rates, and a prime goal of these theories is to predict the values of A and EA for specific chemical systems in terms of quantitative molecular properties.
An important generaltheory that has been adapted for electrode kinetics is the transition state theory, which isalso known as the absolute rate theory or the activated complex theory.Central to this approach is the idea that reactions proceed through a fairly welldefined transition state or activated complex, as shown in Figure 3.1.2. The standard freeenergy change in going from the reactants to the complex is AG*, whereas the complex iselevated above the products by AG\.Let us consider the system of (3.1.1), in which two substances A and В are linked byunimolecular reactions.
First we focus on the special condition in which the entire system—A, B, and all other configurations—is at thermal equilibrium. For this situation, theconcentration of complexes can be calculated from the standard free energies of activationaccording to either of two equilibrium constants:[Complex]_УА'С\^У±ехр(_Афт)[Complex][B](3.1.11)(3.1.12)where C° is the concentration of the standard state (see Section 2.1.5), and y A , y B , and y$are dimensionless activity coefficients.
Normally, we assume that the system is ideal, sothat the activity coefficients approach unity and divide out of (3.1.11) and (3.1.12).The activated complexes decay into either A or В according to a combined rate constant, k', and they can be divided into four fractions: (a) those created from A and reverting back to A,/ A A , (b) those arising from A and decaying to B,/ A B , (c) those created fromВ and decaying to A,/ B A , and (d) those arising from В and reverting back to B,/ B B . Thusthe rate of transforming A into В iskf[A]=fABkf[Complex](3.1.13)* Ъ [ В ] = / В А * ' [Complex](3.1.14)and the rate of transforming В into A isSince we require kf [A] = k\j[B] at equilibrium, / A B and / B A must be the same. In thesimplest version of the theory, both are taken as V2- This assumption implies thatActivated complexReactantProductReaction coordinateFigure 3.1.2 Free energy changesduring a reaction.
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