KINETICS OF ELECTRODE REACTIONS
3.1 REVIEW OF HOMOGENEOUS KINETICS .1 Dynamic Equilibrium
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KINETICS OF ELECTRODE
88 • Chapter 3. Kinetics of Electrode Reactions
At equilibrium, the net conversion rate is zero; hence
£ = * = §* (3.1.5)
^b CA
The kinetic theory therefore predicts a constant concentration ratio at equilibrium, just as thermodynamics does.
Such agreement is required of any kinetic theory. In the limit of equilibrium, the ki- netic equations must collapse to relations of the thermodynamic form; otherwise the ki- netic picture cannot be accurate. Kinetics describe the evolution of mass flow throughout the system, including both the approach to equilibrium and the dynamic maintenance of that state. Thermodynamics describe only equilibrium. Understanding of a system is not even at a crude level unless the kinetic view and the thermodynamic one agree on the properties of the equilibrium state.
On the other hand, thermodynamics provide no information about the mechanism required to maintain equilibrium, whereas kinetics can be used to describe the intricate balance quantitatively. In the example above, equilibrium features nonzero rates of con- version of A to В (and vice versa), but those rates are equal. Sometimes they are called the exchange velocity of the reaction, u0 :
We will see below that the idea of exchange velocity plays an important role in treatments of electrode kinetics.
3.1.2 The Arrhenius Equation and Potential Energy Surfaces (1,2)
It is an experimental fact that most rate constants of solution-phase reactions vary with temperature in a common fashion: nearly always, In к is linear with 1/Г. Arrhenius was first to recognize the generality of this behavior, and he proposed that rate constants be expressed in the form:
(3.1.7) where £д has units of energy. Since the exponential factor is reminiscent of the probabil- ity of using thermal energy to surmount an energy barrier of height E&, that parameter has been known as the activation energy. If the exponential expresses the probability of sur- mounting the barrier, then A must be related to the frequency of attempts on it; thus A is known generally as the frequency factor. As usual, these ideas turn out to be oversimplifi- cations, but they carry an essence of truth and are useful for casting a mental image of the ways in which reactions proceed.
The idea of activation energy has led to pictures of reaction paths in terms of poten- tial energy along a reaction coordinate. An example is shown in Figure 3.1.1. In a simple unimolecular process, such as, the cis-trans isomerization of stilbene, the reaction coordi- nate might be an easily recognized molecular parameter, such as the twist angle about the central double bond in stilbene. In general, the reaction coordinate expresses progress along a favored path on the multidimensional surface describing potential energy as a function of all independent position coordinates in the system. One zone of this surface corresponds to the configuration we call "reactant," and another corresponds to the struc- ture of the "product." Both must occupy minima on the energy surface, because they are the only arrangements possessing a significant lifetime. Even though other configurations are possible, they must lie at higher energies and lack the energy minimum required for
3.1 Review of Homogeneous Kinetics 89
Reactants
Products
Figure 3.1.1 Simple representation of potential energy changes during a
Reaction coordinate reaction.
stability. As the reaction takes place, the coordinates are changed from those of the reac- tant to those of the product. Since the path along the reaction coordinate connects two minima, 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 energy in going from one of the minima to the maximum, which is called the transition state or activated 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 Arrhe- nius equation could be recast as
( 3 1 8 )
We are free also to factor the coefficient A into the product А' cxp(AS*/R), because the exponential involving the standard entropy of activation, Д£*, is a dimensionless con- stant. Then
( 3 1 9 )
or
where AG* is the standard free energy of activation.2 This relation, like (3.1.8), is really an equivalent statement of the Arrhenius equation, (3.1.7), which itself is an empirical generalization of reality. Equations 3.1.8 and 3.1.10 are derived from (3.1.7), but only by the interpretation we apply to the phenomenological constant E&. Nothing we have writ- ten 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 are concentration-dependent. The rate constant is not concentration-dependent in dilute systems; thus the argument that leads to (3.1.10) needs to be developed in the context of a standard state of concentration. The choice of standard state is not critical to the discussion. It simply affects the way in which constants are apportioned in rate expressions. To simplify notation, we omit the superscript "0" from A£*, Д//*, AS*, and AGT, but understand them throughout this book to be referred to the standard state of concentration.
90 Chapter 3. Kinetics of Electrode Reactions
3.1.3 Transition State Theory (1-4)
Many theories of kinetics have been constructed to illuminate the factors controlling reac- tion rates, and a prime goal of these theories is to predict the values of A and EA for spe- cific chemical systems in terms of quantitative molecular properties. An important general theory that has been adapted for electrode kinetics is the transition state theory, which is also known as the absolute rate theory or the activated complex theory.
Central to this approach is the idea that reactions proceed through a fairly well- defined transition state or activated complex, as shown in Figure 3.1.2. The standard free energy change in going from the reactants to the complex is AG*, whereas the complex is elevated above the products by AG\.
Let us consider the system of (3.1.1), in which two substances A and В are linked by unimolecular reactions. First we focus on the special condition in which the entire sys- tem—A, B, and all other configurations—is at thermal equilibrium. For this situation, the concentration of complexes can be calculated from the standard free energies of activation according 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 yA, yB, and y$
are dimensionless activity coefficients. Normally, we assume that the system is ideal, so that 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 con- stant, k', and they can be divided into four fractions: (a) those created from A and revert- ing back to A,/A A, (b) those arising from A and decaying to B,/A B, (c) those created from В and decaying to A,/B A, and (d) those arising from В and reverting back to B,/B B. Thus the rate of transforming A into В is
kf[A]=fABkf [Complex]
and the rate of transforming В into A is
* Ъ [ В ] = / В А * ' [Complex]
(3.1.13)
(3.1.14) Since we require kf [A] = k\j[B] at equilibrium, /A B and /B A must be the same. In the simplest version of the theory, both are taken as V2- This assumption implies that
Activated complex
Reactant
Product
Reaction coordinate
Figure 3.1.2 Free energy changes during a reaction. The activated complex (or transition state) is the configuration of maximum free energy.
3.2 Essentials of Electrode Reactions ^i 91 /AA = /вв ~ 0 »t n u s complexes are not considered as reverting to the source state. Instead, any system reaching the activated configuration is transmitted with unit efficiency into the product opposite the source. In a more flexible version, the fractions /дв and /B A are equated to к/2, where к, the transmission coefficient, can take a value from zero to unity.
Substitution for the concentration of the complex from (3.1.11) and (3.1.12) into (3.1.13) and (3.1.14), respectively, leads to the rate constants:
Statistical mechanics can be used to predict кк'/2. In general, that quantity depends on the shape of the energy surface in the region of the complex, but for simple cases k' can be shown to be 26T/h, where, 4 and h are the Boltzmann and Planck constants. Thus the rate constants (equations 3.1.15 and 3.1.16) might both be expressed in the form:
(3.1.17) which is the equation most frequently seen for calculating rate constants by the transition state theory.
To reach (3.1.17), we considered only a system at equilibrium. It is important to note now that the rate constant for an elementary process is fixed for a given temperature and pressure and does not depend on the reactant and product concentrations. Equation 3.1.17 is therefore a general expression. If it holds at equilibrium, it will hold away from equilib- rium. The assumption of equilibrium, though useful in the derivation, does not constrain the equation's range of validity.3