Keith J. Laidler University of Ottawa Ottawa 2, Canada
The Kinetics of Electrode Processes
the surface simply a unimolecular layer of idns of opposite sign to the charge existing on the solid; this model is illustrated in Figure la, which shows that the electrical potential changes sharply from its value in the solid to that at the center of the ionic layer.
A s a result of much experimental and theoretical study there has now been gained a considerable insight into the kinetics and mechanisms of reactions occurring at electrodes. Aside from it,s intrinsic scient,ific interest, this topic is of great technical importance, since it is related to many industrial processes and to the operat,ion of practical devices such as storage batteries and fuel cells. However, in spite of the importance of this area of research, undergraduate textbooks of physical chemistry provide very little information about it. Since, in addition, there have been few if any reviews of the subject at the undergraduate level, the topic seems to be largely ignored in the teaching of physical chemistry. Conway and Salomon (1) have recently outliued the areas of electrochemistry that they consider should he emphasized in an undergraduate course in physical chemist.ry, and have stressed the import,ance of electrochemical lcinet,ics, of which they give a brief account,. Certain aspects of the subject have also been dealt wit,h by Parsons (%). It is the object of the present article to give a more detailed account of t,he kinetics and mechanisms of electrode processes, at a level suitable for inclusion in a first or second course in general physical chemistry. For more advanced account,^ of the present topic the reader is referred to Potter's "Electrochemist~ry" (3) and to Rortum's "Treat,ise on Electrochemistry" (4). Still more detailed t,reatments are given in books by Vetter ( 5 ) , Conway (6), and Delahay (7) and in reviews by Conway (8) and by Boclcris (9). Readers who refer t,o Chapter X of Glasstone, Laidler, and Eyring's "The Theory of Rate Processes" (10) should note that alt,hough the basic theory given there is substantially correct, the application to experimental data is outdated so that some of the conclusions drawn (as to the rate-determining step for certain systems) are now knovn to be incorrect (see, for example, the footnote toward the end of the present art,icle).
(0 1
Figure 1. Various models for the structure of the electrical double layer, ~ h o x i n gthe variations of electrical potential with distance from the surface: lo) Helmholtz model; Ibl Gouy-Chopmon (diffuse double layer1 model; (c) Stern's combination of the Helmholtz ond Gouy pmposols; (d) Grohome model; Ie)model of Devanothon, Bockns, and Muller.
The Electrical Double Layer
The processes occurring at electrodes are closely related to the structure of the region of the solution that is near to the electrode; the rate-determining step in an electrode process involves substances present in t,his region, which has come to be known as the "double layer." Ideas about the structure of the double layer have developed st,eadily over the past ninety years, and have been closely related to advances in other fields, such as the treatments of the ionic atmosphere given by Debye and Hiickel (11). The first and simplest theory of the double layer was that of Helmholtz (I!?), who considered that there is at 600 / Journal o f Chemical Education
fcl
'
Gouy (IS) and, independently, Chapman (Id), pointed out that the Helmholtz model was unsatisfactory in neglecting the Boltzmann distribution of the ions in a varying potential field. I t is of interest, in this connection, that ten years before the development of t,he Dehye-Huckel theory (11) for the distribution of ions round a central ion, Chapman had worked out an analogous treatment for the distribution of ions in the neighborhood of a charged solid. As shown schematically in Figure l b , this Gouy-Chapman treatment leads to an ionic-atmosphere type of distribution, with the potential falling more gradually through the diffuse layer formed. A double-layer model which essentially combines the Helmholtz and Gouy-Chapman ideas was proposed by Stern (15), who recognized that a satisfactory treatment must take into account the finite size of t,he
ions involved and the specific chemidorptive interactions between the ions and the surface. His model is represented schematically in Figure lc. The surface is shown bearing a negative charge relative to the solution, and there is a layer of positive ions held a t the surface by chemisorptive and electrostatic attractions. The region from the surface to the center of this layer of ions may be referred to as the Helmholtz layer; there is a sharp potential drop from J., t o h. Outside the Helmholtz layer there is a Gouy-Chapman or diffuse layer, over which the potential drops more gently from to J.,, the potential of the bulk of the solution. Subsequent to Stern's work there have been a number of further proposals which refine the model in various details. Thus Grahame (16) concluded that the effects of adsorbed anions on the potentials of mercury electrodes could not be satisfactorily interpreted on the basis of the Stern model. He therefore extended it in the manner indicated in Figure Id. There is a potential drop from to G2 from the metal to the center of a layer of specifically adsorbed ions. There is a further drop to J.L at the inner limit of the diffuse layer, where there is a layer of ions which are separated from the surface by a layer of solvent molecules and are held by electrostatic but not chemisorptive forces. Finally there is a drop from to J., over the diffuselayer. The place a t which the potential is J.%is referred to as t.he inner Helmholtz layer, that at which it is as the outer Helmholtz layer (cf. Fig. lc). However, even this improvement is not sufficient to explain all of the experimental results. Grahame (17) pointed out that the capacity of the double layer at mercury is largely independent of the sizes of alkali and alkaline-earth cations; this is inconsistent with the Stern model, according to which the various cations would have different distances of closest approach to the metal. Furthermore, the results on the adsorption of anions, X-, on mercury are not readily explicable in terms of the bond strengths between the mercury surface and X-; instead the effects seem to be related to the degree of hydration of the X- ions. In order to explain these and other results Devanathan, Bockris, and Miiller (18) proposed the model shown in Figure Id. According to this, there exists at the surface a layer of strongly adsorbed solvent molecules, held by charge-dipole forces. Specifically adsorbed anions are considered to be capable of penetrating this solvent layer. The outer plane of this solvent layer is referred to as the Helmholtz plane, and the potential is +x. Adsorbed cations with their primary hydration shells are regarded as remaining outside this plane of solvent molecules; the plane through the center of this layer of cations is referred to as the Gouy plane, and its potential is +I. Outside this plane there is a further drop of potential from +I to J.,over the diiuse layer. It is outside the scope of the present article to deal with elect~,okinetieeffects, which relate to the relative movement of layers of solid or liquid arising from potential differences, but a brief reference should be made to the zeta-potential. If two planes are a t different electrical potentials and are able t o slide past one another, such movement will occur when a potential field is applied. The speeds of such motions
+,
+,
in the case of a solid-liquid interface therefore depend upon the magnitude of the potential difference between the outer part of the surface layer that is fixed with respect to the solid, and the bulk of the solution. This potential difference is the zeta-potential, $. It has no particular significance with respect to the Helmholtz model of the double layer. For the other models it is the potential drop $1 across the diffuse part of the layer. It is to be noted that the sign of the f potential need not be the same as the sign of the overall potential change - J.,, since the potential at the edge of the diffuse layer may differ from that of the solid surface, owing to absorption effects.
+,
Distribution of Species at Electrodes
An important matter that arises in connection with the kinet,ics of electrode processes is the manner in which species present in solution distribute themselves in the neighborhood of the electrode. There are two main topics to consider 1) The concentrations of solute species at the outer Helmholtz, or Gouy, plane; this is the plane which hounds the diffuse layer. 2) The concentrations of adsorbed species on the electrode surface.
The first problem will be discussed with reference to the discharge of hydrogen ions at a negatively charged electrode, the overall process being eH+ '/%Hz. AS will be seen, for many systems there is rapid establishment of equilibrium between H + ions' in the bulk solution and H + ions in the Gouy plane. Attention is therefore concentrated on these H + ions in the Gouy plane, as the reactants in the discharge process, but it is necessary to relate their concentration, [H+Ic, to the concentration [H+] in the bulk solution. Their distribution will, in fact, be determined by the electrical potential +I at the Gouy plane (with reference to the electrical potential in the bulk solution, J.,, which is taken as zero). By the Boltzmann principle
-
+
where F is the faraday; 4 9 is the electrical work required to transport a unit charge from the bulk solution t o the Gouy plane. The magnitude of the potential +I is thereforc an important matter in the theory. If the H + ions are the only cations present in the solution a treatment similar to that of Debye and Hiickel leads to the conclusion (14,19-21) that $1 will vary with [H+] as
where 40 is a constant. When this is the situation it follows from eqns. (1) and (2) that [H'], will be independent of [H+]. When other cations are present the situation is by no means simple. The concentration [H+] in eqn. (2) should now be replaced by something like an ionic-strength term but with only cations included, i.e., by --
'For convenience, hydrogen ions will be written simply as H+;it is to he understood that they are always strongly attached to surrounding water molecules. Volume 47, Number
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for all cations. However, this is an over-simplificat,ion; the anions have some effect in determining 41, since they distribute themselves in the diffuse double layer. Fortunat,ely, the exact expression for 4, is not of great importance since it is common in electrode kinetic work to operate at a relatively high concentration of foreign and inert ions, the effect of which is to maintain 41 at a constant, even if unknown, value. For example, in pH studies the concentration of foreign ions may be held a t a much higher value than that of the H + ions, so that the pH can be varied over a wide range without 41being affected. The magnitude of [H+]also, of course, affects the value of J.,,in the manner to be considered later (cf. eqn. (12), which refers to the situat,ion when the electrode is operating reversibly). However, it is important to note that, according to the theory of the diffuse double layer, the magnitude of 4, is not affected by the potential applied to the electrode. Potentials that are applied in excess of t,he reversible potential (i.e., the overvoltage) therefore operate only across the region between the electrode and the Gouy plane, not across the diffuse layer. The second topic pertaining t,o the distribution of species at the electrode relates to adsorption at the electrode. I n some discharge mechanisms the ratedetermining step involves species such as adsorbed hydrogen atoms, or occurs at surface sites that are not occupied by solute species, and in either case it is necessary to make use of an adsorption isotherm. The simplest of these is the Langmuir adsorption isotherm, according to which
CONCENTRATION Figure 2. The variot;ons with concentration of the fracson 0 of 0 rurfoce covered by on adsorbed species, as given by the Langmuir and Temkin isotherms.
principles discussed here are applicable, with very little modification, to other electrode processes, including the cathodic deposition of metals and the evolution of oxygen gas at anodes. If an electrode is maintained in an aqueous acidic solution at its reversible potent,ial the overall rate of the above reaction is equal to the rate of the reverse reaction; there is therefore no net flow of current, and no net chemical change. If the electrode is made slightly more negative than its reversible potential the e '/nHz will occur slightly more process H + rapidly than the reverse reaction; there will therefore be a net flow of current. Conversely, if a more positive potential is applied to the electrode the process '/*HZ + H + e- will occur more rapidly than the reverse. If the occurrence of reaction in the direction H+ e 1/2H2is talcen arbitrarily to correspond to a positive flow of current, it follows that the current i decreases with an increase in the positive potential V of the electrode. Experimentally (and theoretically, as will be secn) the variation of i with V is an exponential one, as shown in Figure 3. The absolute magnitude
+
+
+
Here [XIadais the concentration of adsorbed species and 9 the fraction of the surface covered by it; [XI is the concentration of the solute X in solution and lc and K are constants. This isotherm involves the assumption that there are no interactions between adsorbed species and probably leads to some error a t intermediate coverages. An empirical isotherm that is consistent with intermolecular interactions has been proposed by Temlcin ($9); when applied to the adsorption of hydrogen ions this isotherm leads to the conclusion that at intermediate coverages 0 varies logarithmically with [XI
-
+
CURRENT, i
EXPONENTL
REGION
where A and R are constants. Figure 2 shows in a schematic way the different variations of 8 with [XI to be expected from the two isotherms. Elementary Steps in the Cathodic Evolution of Hydrogen
WTENTIAL.V DEPENDENCE
Much of the work on overvoltage and the mechanisms of electrode processes has been done on the evolution of gaseous hydrogen at a cathode; if the solution is acidic the overall process is
Variation of current i with potential V, for discharge ot o cathode. The current is token to b e positive when discharge occurs, and is therefore deereared by a n increase in t h e potential of the cathode, ond is increased when t h e cathode is made more negative. Figure 3.
The present article is concerned exclusively with the various mechanisms for this process. Many of the 602 / Journal of Chemical Education
with respect to the of the reversible potential, solution cannot he determined; instead a value Vo is used which is related arbitrarily to that of a standard electrode. If the potential is displaced slightly from V oto a value V the current i varies linearly with V ; at larger V - V ovalues the variation is much stronger. The quantity V - V ois known as the overvoltage, 7, of the system a t that particular current. Sometimes the slope of the curve is very small in the region of Vo; in such cases a suhst,antial overvoltage has to be applied in order for any significant amount of current to flow. In the case of hydrogen evolution, for example, an overvoltage -7 of as much as 1 volt may be necessary before bubbles of hydrogen can be observed on the electrode. The way in which i changes with q depends upon the speed with which the electrode process occurs; if i t is slow the slopes in Figure 3 will be small. Much of the research effort that goes into the study of overvoltage is concerned with elucidating the nature and relative rates of the elementary processes that occur. The situation is quite complicated, and varies considerably with the nature of the electrode, the concentration of the electrolyte and the magnitude of the overvoltage applied. There is sometimes a simplification if one particular process is rate determining; even when this is the case, however, pre-equilibria have to be taken into account. The various processes that may occur at the cathode, when hydrogen is being evolved, are as follows 1) The transport of H + ions from the bulk of the solution to the electrode double layer. This process is sometimes rate-determining, but very often it can be shown not to be by the following experimental results (a) The ?-i relationship frequently depends to a considerable extent on the nature of the electrode surface. (b) The activation energy associated with overvoltage (see later) is usually much larger than that (2-3 kcal/mole (10)) associated with the movement of a proton in aqueous solution. However, although the rate with which H + ions get to the surface is irrelevant to overvoltage, the preequilibrium between bulk H + ions and those in the double layer is of great importance, since it controls the concentration of H + ions in the double layer. 2) The interaction between a hydrated H + ion in the double layer and an electron in the metal, with the formation of a chemisorbed H atom. There is good reason to believe that this process is frequently ratedetermining. It is reasonable that this should he so, since the breaking of a chemical bond (between the H + ion and the attached water molecule) is involved, and this will give rise to a substantial activation energy. The dependence of the rate on the nature of the metal is also explained if this process is slow. The 7-i relationship to he expected if this process is rate-determining is considered later (cj. eqn. (21)). 3) There are several ways in which chemisorbed H atoms can become H, molecules; one is by direct recombination, 2H -t Hz. If this process is ratedetermining, which it probably is on some surfaces, the overall rate will depend not only on the rate of this
recombination, but also on the pre-equilibria involving processes (1) and (2) above. Since the recombination reaction does not involve an electron transfer its rate is not affected by the potential. However, the concentration of H atoms is affected by the rate of the preceding reaction, which produces the adsorbed H atoms and which does involve an electron transfer. Consequently, although the rate-determining step is independent of V , the overall process is affected by V ; the exact relationship is considered later (eqn. (35)). 4) Chemisorbed H atoms might also be removed by reaction with hydrated H + ions in the double layer e-+M-H+H+-M+H2
Since this process involves an electron transfer, its rate is affected by V . Since also the concentration of M-H is affected by V (through the pre-equilibrium), it is apparent that V now has a dual effect on the magnitude of the current; the i-? relationship is derived below (eqn. (41)). 5) Finally, the hydrogen molecules must come together and form bubbles of gas, which are evolved. There is little evidence that this process ever controls the rate; it is presumably fairly fast and will require no activation energy. The following are therefore possible mechanisms for the evolution of hydrogen gas from an aqueous acidic solution Mechanism I
Either reaction may be rate controlling or, more generally, the overall rate may involve all three rate constants. The suggestion that the first step is slow was first made by Erdey-Gruz and Volmer (23); a slow second step is the main feature of the mechanism of Tafel (84). Mechanism 11
k,'
M-H+Hf+M+H2
Again, either reaction may he rate-determining. In the former case the kinetics are the same as with Mechanism I. The second possibility was explored by Heyrovslry (26). Rate Equations for Mechanism I
We consider first Mechanism I for the case in which the initial step, the transfer of a proton from the Gouy plane to the surface, is rate controlling. Figure 4 gives a representation of the situation from the standpoint of the double-layer models proposed by Grahame (16) and by Devauathan, et al. (18). However, the exact model employed is not crucial to the derivation of the kinetic equation. The main points are 1 ) There is a diffuse double layer separating the bulk solution from the Gouy plane where there exists B layer of hydrated protons at 8. potential of dr relative to the solution. (It must be noted that although d, is for oonvenience shown as posit,ive in the diagram, it is usudly negative.) 2) On the surface there is an adsorbed layer of water molecules. Volume 47, Number 9, September 1970
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It is assumed that the equilibrium between H + ions in the Gouy plane and those in the bulk solution is rapidly established, so that the slow process is simply the transfer of H + from the Gouy plane to the surface. The diagram shows an H + ion in the initial state and the adsorbed H atom in the final state; it also shows the variation of potential from the H atom at the metal to the bulk solution. As the H+ ion travels from the initial to the final state there is an increase of potential of h - 41 (in a hydrogen evolution experiment this is numerically a decrease since $m - $1 is negative, i.e., the metal is negatively charged with respect t o the Gouy plane).
The rate constant for the process can therefore be expressed as
where A is the frequency factor; in terms of activated complex theory (10, B6), A could be expressed as (kT/h) exp (-AS*/RT) where AS* is the entropy of activation, k is the Boltzmann constant and h is Planck's constant (the transmission coefficient is presumably close to unity for a reaction of this kind). The rate of the process will be this rate constant multiplied by the concentration [H+Io of protons in the Gouy plane and also by the fraction of sites on the surface which are bare and therefore able to accept the hydrogen atom. If the fraction of surface covered by H atoms and other adsorbed species (but not including solvent molecules, which are assumed to be easily replaceable by H atoms) is 0, the fraction uncovered is 1 - .'6 The rate of proton transfer can therefore be written as
We must also have an expression for the rate of reaction in the reverse direction. The activation energy in this direction is raised by (1 - @)(+I- &)F and is therefore equal to E*-I + (1 - I3)(41 - &)F (cf. Fig. 4). The rate constant is thus k,=
POTENTIAL ENERGY
A _ , ~ - [ E - ~ * + ( ~ - O ) ( ~ ~ - - ~ ~ ) F I I R T (7)
The reactant is now the adsorbed H atom, of concentration [HI.; the rate is thus
Figure 4. The upper diagram rhowr the initial and Rnal +des in the cdhodic discharge of a hydrogen ion in the Gouy plane, and the voriatim of electrical polential. The potenlid of the bulk solution, Jla, is taken arbitrarily as zero. The potentials 61,ond $Z ore normally negative with respect to the bulk solution when discharge is taking place, and 61is The lower diogrorn rhowr the vmriolions in potential energy with dirtmce from h e rurfoce. The dashed cune for M-H is for = 6,;the firm c u n e applies when # 6,,and is below the doshed curve by h e positive ornovnt Idr +-IF.
,+,
(It is assumed that there is always a water molecule available to accept the proton, so that there is no need to multiply this by the fraction of the Gouy plane occupied by water molecules.) Suppose now that the electrode is operating at its reversible absolute potential firno; there is then no net reaction, so that vl = 0-1 or
+, +,
-
+,
The lower part of Figure 4 shows a potential energy diagmm for the transfer process. The dashed curve at the left shows the potential energy variation as the distance At-H is altered. The curve a t the right shows the potential energy changes as the proton is moved from its equilibrium position in the Gouy plane. Taken together, the left-hand dashed curve and the right-hand curve show schematically the relative positions in the absence of the electrical potential difference. However, because of the variation in electrical potential the left-hand curve is lowered, relative to the right-hand one, by the amount (41 - &)F, where F is the faraday. I n the absence of any potential difference 41 - J., the activation energy for the proton transfer would be El*; in its presence it is less than this, but not by the full amount (41 - $,)F. The diagram shows that the lowering of the activation energy is actually by an amount B(4, - $,)P, wl?ere 13 is a fraction; if the curves cross in a symmetrlcal fashion the value of I3 would be 604
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This equation rearranges to
However, the concentration [H+], is related to that in the bulk solution [H+], by eqn. (1); eqn. (10) thus becomes [H+l -
-
[HI.
A-3 -~E-~*-E*)/RT~~F/RT A1(1 - 0 )
The variation of
(11)
with [H+]is therefore as follows
This equation is equivalent to the Nernst equation according to which the hydrogen electrode potential relative to an arbitrary standard electrode is given by
$,O,
The conclusion is that at the reversible potential when the reactions occur at equal rates in the two
directions, the rates are given by eqns. (6) and (8) with $, set equal to $,Q. The currents are those expressions multiplied by F; these equal and opposite currents are called the exchange currents, and are seen from eqns. (6) and (8) to be given by
example, for the deposition of hydrogen on palladium and smooth platinum electrodes from acid solutions b is about one-quarter of 0.12, viz. 0.03. It will be seen that this can be interpreted on the basis of Tafel's mechanism that the slow step is the combination of hydrogen atoms (cf. eqn. (35)). Influence of pH on Overvoltage
Suppose now that an additional positive potential v( = $m - Jlmo)is applied to the electrode; this potential is known as the ouemoltage. As noted early, there is reason to believe that this overvoltage does not affect the potential @I; that is, it only affects the potential drop between the electrode and the Gouy layer. This being so, it has the effect of decreasing the deposition rate by the factor exp (-BqF/RT), and of increasing the rate of the reverse reaction by the factor exp [(I - b)?F/RT]. The new currents are therefore i, =
i,e-P+"/RT
i,= ioe(l-P)nF/RT
(16) (17)
The net current i (taken as positive if there is a net evolution of HZ)is thus This relationship between the current i and the overvoltage q takes simple forms in two limiting cases 1) If 7 is close t,o zero the exponentials in eqn. (18) may be expanded and only t,he first terms accepted
Insight into the mechanism of the discharge of hydrogen at cathodes is provided by studies in which the 7 versus loglo i relationship is studied over a range of pH values. There are two conditions under which this is conveniently done. In the first place, one can work with aqueous solutions containing nothing but an acid, the concentration of which is varied. Alternatively, one can vary the pH in a solution containing an excess of a chemically inert salt; this procedure has the advantage of maintaining @I constant while the pH is being varied. If nothing but acid is present the value of 41 is controlled entirely by the concentration of H + ions, through the relationship (1). The current when the exponential eqn. (21) is obeyed is obtained by combining eqns. (14) and (21), and is
which may be simplified to However, in view of eqn. (I), the quantity [H+]e-+lF/RT is mdependent . of [H+], as was noted earlier; this quantity is in fact the concentration of H + ions in the Gouy layer. Moreover the potential difference JlmO- @I is independent of [H+],since both J.,Q and 4, vary in the same manner with [H+] (Jlmoby eqn. (12) and @I by eqn. (1)). I t follows that for this case of discharge from a pure acid solution the 7 versus log,, i relationship is pH-independent. The situation is different if there is an excess of inert cations present, in which case @I is maintained constant. Equation (25) may then be written as i = eonst. [H+]e-B(+d + n)F/RT (26) '
Under these conditions there is a linear dependence of i on 7 (cj. Fig. 3). By the use of this relationship one can obtain i, by making measurcments of i in the region of the reversible potential. 2) If 7 has a larger positive or negative value one of the exponentials in eqn. (18) becomes negligible in comparison with the other. Suppose, for example, that q is made large and negative; the equation becomes
the terms in 41 now being included in the constant. Taking logarithms h g , i~ = c a s t
+ log10 [A+]-
Insertion of eqn. (12) for
leads to
where a is a constant. This equation is of the form where b = 2.303RT/BF. Plots of 7 against loglo i should therefore be linear, and this has been verified for a number of systems. Equation (23) is linown as the Tafel equation, after Tafel, who first arrived at it empirically (24). The slope b is linown as the Tafel slope. If fl = '/2 the magnitude of 2.303 RT/PF in water a t 25'C is about 0.12. For a number of electrode systems (e.g., mercury electrodes in acid solution) the slopes of the 7 versus log,, i plots have indeed this value. Other systems obey the Tafel equation, but give slopes which differ significantly from 0.12. For
A plot of q against. pH, a t constant current i, should therefore yield a straight line of slope equal to
If P = the magnitude of this in water at 25°C is -0.06. Experiments at constant salt concentration, with a mercury electrode, were described by Frumkin (27), who found the predicted behavior to be obeyed in acid solution. I n alkaline solution, however, the behavior is different, as is shown schen~atically in Figure 5. I n alkaline solutions the negative potentials Volume 47, Number 9, September 1970
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Insertion of the expressions in eqns. (16) and (17) for il and i-I leads to
If the surface is sparsely covered the rate of combination will be proportional to [M-HI2 and is therefore vl = b[M-HI"
hklIH+]%-W/RT
(33)
Since the combination of atoms does not involve charge transfer the rate constant kz does not involve q. The current i is proportional to v2, and therefore, at constant pH
Figure 5. The variation of overvoltage q with pH, d constant current, for the mechanism in which proton trmnrfer to the rvdace is mte-determining. The left-hand limb shows the behavior in w i d solution when Hf is tronrferred b y a hydrated proton in the Gouy layer; the right-hand limb, in more alkaline rolution, corresponds to transfer of H+from o water molerule in tho Govy layer.
required to produce a given current are lower than predicted by eqn. (28), and this suggests that a new mechanism takes over when the H + ion concentration in the Gouy plane is too low for the process eH+ A4 -+ MH to be efficient. An obvious possibility is that the water molecules are now providing the protons, the mechanism being
+ +
e-+H%O+M-M-H
f OH-
If this is the case it can be seen by analogy with eqn. (25) that the rate equation will now he i = c a s t e-IBIlno+r-di)fl/RT
= east
BnF - B logu [Ht1 - 2.303~
(30)
The slope of the plot of q against pH a t constant i is now 2.303 RT F
(cf. Fig. 5). This corresponds to the behavior obsewed, and it is therefore concluded that in alkaline solution the rate-controlling step does indeed become the transfer of a proton from a water m o l e ~ u l e . ~ Alternative Mechanisms
These two mechanisms in which proton transfer from hydrated H + or from HzO in the double layer is rate-determining are by no means the only ones that are observed. Another situation that commonly arises is that the combination of adsorbed H atoms on the surface is rate-controlling. If the first reaction in Mechanism I is rapid in both directions the concentration of adsorbed H atoms, i.e., of M-H species, is given by
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The slope of the plot of q against pH, at oonstant i, is therefore
--2303 RT = -0.06 F
(29)
where the constant includes the concentration of water molecules in the Gouy layer. The potential $,O varies with [H+] through eqn. (12) and +I is held constant; then log,, i
This is again of the Tafel form (indeed it was on the basis of this mechanism that Tafel first derived the equation), but the Tafel slope b is now 2.303 RTI2F. This is about one-quarter of that obtained when the proton transfer is rate-determining, its value for water a t 25°C being -0.03. Such values have in fact been obtained for certain systems, such as platinum and palladium electrodes in acid solution. The pH dependence for this case is easily deduced from eqn. (33). The oonstant kz is independent of both q and pH; thus
for water at 25'C
Another possible mechanism is Mechanism 11, given earlier. If the first step is rate-controlling the behavior is, of course, the same as discussed previously for Mechanism I. Alternatively, the slow step may he the second stage, the interaction between the H + ion in the double layer and the adsorbed atom, as suggested by Heyrovsky (85). The overall rate is now where ki' is the rate constant for the interaction hetween Rf-H and H+, and [M-HI is given by eqn. (32). Since the slow step now involves a protontransfer it will he affected by the potential drop tm0 q - 61,so that k,' has the form
+
Here Aa' is the frequency factor, E2* the energy of activation in the absence of the potential drop, and 8' Were a specific correction to "The Theory of Rate Processes" ((lo),cf. refs. (28) and (39)) should be made. It was there wnoluded from the pH-dependence of the 7 versus log i relationship that the slow step must be proton transfer from a water molecule, even in acid solution. However, it was not then clear that the pH independence only applies when no salts are present, and that this follows from eqn. (25). The work of Frumkin ($7) and much later work has shown that when 0, is held wnstant by excess salt the behavior is as shown in Figure 5; this clearly demonstrates that proton transfer is from hydrated H + in acid solution, and from HIO only in alkdine solution.
the symmetry factor for this process. Insertion of eqns. (38), (32), and (1) into (37) leads to i = const. [H+]ze - + ! P / R T G - V F / R T e-5'nF/RT @'&P/RT
Literature Cited
(39)
Terms independent of q and [ H f ] have been put into the constant. The relationship between i and v given by this equation, at constant pH, is then
This is again of the Tafel form; the Tafel slope is now 2.303 R T / ( l p')F;
