A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 27
Текст из файла (страница 27)
The activatedcomplex (or transition state) is theconfiguration of maximum freeenergy.3.2 Essentials of Electrode Reactions ^i 91=tnus/AA /вв ~ 0 »complexes are not considered as reverting to the source state. Instead,any system reaching the activated configuration is transmitted with unit efficiency into theproduct opposite the source.
In a more flexible version, the fractions /дв and / B A areequated 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 theshape of the energy surface in the region of the complex, but for simple cases k' can beshown to be 26T/h, where, 4 and h are the Boltzmann and Planck constants. Thus the rateconstants (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 transitionstate theory.To reach (3.1.17), we considered only a system at equilibrium.
It is important to notenow that the rate constant for an elementary process is fixed for a given temperature andpressure and does not depend on the reactant and product concentrations. Equation 3.1.17is therefore a general expression. If it holds at equilibrium, it will hold away from equilibrium. The assumption of equilibrium, though useful in the derivation, does not constrainthe equation's range of validity.33.2 ESSENTIALS OF ELECTRODE REACTIONS (6-14)We noted above that an accurate kinetic picture of any dynamic process must yield anequation of the thermodynamic form in the limit of equilibrium.
For an electrode reaction,equilibrium is characterized by the Nernst equation, which links the electrode potential tothe bulk concentrations of the participants. In the general case:O + ne^R(3.2.1)this equation is(3.2.2)where CQ and CR are the bulk concentrations, and E° is the formal potential. Any validtheory of electrode kinetics must predict this result for corresponding conditions.3Note that 4T/h has units of s" 1 and that the exponential is dimensionless.
Thus, the expression in (3.1.17) isdimensionally correct for a first-order rate constant. For a second-order reaction, the equilibrium corresponding to(3.1.11) would have the concentrations of two reactants in the denominator on the left side and the activitycoefficient for each of those species divided by the standard-state concentration, C°, in the numerator on the right.Thus, C° no longer divides out altogether and is carried to the first power into the denominator of the finalexpression. Since it normally has a unit value (usually 1 M~l), its presence has no effect numerically, but it doesdimensionally.
The overall result is to create a prefactor having a numeric value equal to 6T/h but having units ofM~l s" 1 , as required. This point is often omitted in applications of transition state theory to processes morecomplicated than unimolecular decay. See Section 2.1.5 and reference 5.92Chapter 3. Kinetics of Electrode ReactionsWe also require that the theory explain the observed dependence of current on potential under various circumstances.
In Chapter 1, we saw that current is often limited whollyor partially by the rate at which the electroreactants are transported to the electrode surface. This kind of limitation does not concern a theory of interfacial kinetics. More to thepoint is the case of low current and efficient stirring, in which mass transport is not a factor determining the current. Instead, it is controlled by interfacial dynamics.
Early studiesof such systems showed that the current is often related exponentially to the overpotentialг]. That is,фi - а' е '(3.2.3)or, as given by Tafel in 1905,' г] = a + b log / '(3.2.4)A successful model of electrode kinetics must explain the frequent validity of (3.2.4),which is known as the Tafel equation.Let us begin by considering that reaction (3.2.1) has forward and backward paths asshown.
The forward component proceeds at a rate, Vf, that must be proportional to the surface concentration of O. We express the concentration at distance x from the surface andat time t as Co(x, t)\ hence the surface concentration is CQ(0, i). The constant of proportionality linking the forward reaction rate to CQ(0, i) is the rate constant kf.vf = kfCo(0, t) = - ^(3.2.5)Since the forward reaction is a reduction, there is a cathodic current, /c, proportional toLikewise, we have for the backward reaction(3-2.6)where /a is the anodic component to the total current.
Thus the net reaction rate is"net = vf-vb= kfCo(0, t) - kbCR(0, t)=-^(3.2.7)and we have overalli = i c - i a = nFA[k{ C o (0, t) - kbCR(0, t)](3.2.8)Note that heterogeneous reactions are described differently than homogeneous ones.For example, reaction velocities in heterogeneous systems refer to unit interfacial area;hence they have units of mol s" 1 cm~ 2 .
Thus heterogeneous rate constants must carryunits of cm/s, if the concentrations on which they operate are expressed in mol/cm3. Sincethe interface can respond only to its immediate surroundings, the concentrations enteringrate expressions are always surface concentrations, which may differ from those of thebulk solution.3.3 BUTLER-VOLMER MODEL OF ELECTRODE KINETICS(9,11,12,15,16)Experience demonstrates that the potential of an electrode strongly affects the kinetics ofreactions occurring on its surface. Hydrogen evolves rapidly at some potentials, but not atothers. Copper dissolves from a metallic sample in a clearly defined potential range; yetthe metal is stable outside that range.
And so it is for all faradaic processes. Because theinterfacial potential difference can be used to control reactivity, we want to be able to pre-3.3 Butler-Volmer Model of Electrode Kinetics93diet the precise way in which k{ and kb depend on potential. In this section, we will develop a predictive model based purely on classical concepts. Even though it has significant limitations, it is very widely used in the electrochemical literature and must beunderstood by any student of the field. Section 3.6 will yield more modern models basedon a microscopic view of electron transfer.3.3.1Effects of Potential on Energy BarriersWe saw in Section 3.1 that reactions can be visualized in terms of progress along a reaction coordinate connecting a reactant configuration to a product configuration on an energy surface.
This idea applies to electrode reactions too, but the shape of the surfaceturns out to be a function of electrode potential.One can see the effect easily by considering the reactionNa+ + e^Na(Hg)(3.3.1)where Na+ is dissolved in acetonitrile or dimethyIformamide.
We can take the reaction coordinate as the distance of the sodium nucleus from the interface; then thefree energy profile along the reaction coordinate would resemble Figure 3.3.1a. To-^~ OxidationReduction -(a)Na(Hg)OxidationP(b)Na(Hg)Reduction -«s-(c)IAmalgamSolution •Reaction coordinateFigure 3.3.1 Simplerepresentation of standard freeenergy changes during afaradaic process, (a) At apotential corresponding toequilibrium, (b) At a morepositive potential than theequilibrium value, (c) At amore negative potential thanthe equilibrium value.94Chapter 3.
Kinetics of Electrode Reactionsthe right, we identify N a + + e. This configuration has an energy that depends littleon the nuclear position in solution, unless the electrode is approached so closely thatthe ion must be partially or wholly desolvated. To the left, the configuration corresponds to a sodium atom dissolved in mercury. Within the mercury phase, the energydepends only slightly on position, but if the atom leaves the interior, its energy risesas the favorable mercury-sodium interaction is lost. The curves corresponding tothese reactant and product configurations intersect at the transition state, and theheights of the barriers to oxidation and reduction determine their relative rates.When the rates are equal, as in Figure 3.3.1a, the system is at equilibrium, and thepotential is Eeq.Now suppose the potential is changed to a more positive value. The main effect is tolower the energy of the "reactant" electron; hence the curve corresponding to Na + + edrops with respect to that corresponding to Na(Hg), and the situation resembles that ofFigure 3.3.Ib.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
