SIDNEYBARNARTT
412
Modified General Theory of Charge-Transfer Electrode Kinetics
by Sidney Barnartt Edgar C. Bain Labordory for Fundamental Research, Unaed States Steel Corporation, Research Center, Monroeville, Pennsylvania (Received July 19, 1966)
The kinetic theory of electrode reactions contained several unsatisfactory aspects. The modified general equations derived here largely eliminate these drawbacks. A rigorous thermodynamic treatment of the activation process shows that the activity of the activated complex at the electrode in its standard state is a definite quantity not open to arbitrary selection. The activity coefficient of the complex can be eliminated under specified conditions. The modified rate equations yield energies of activation which are determined primarily by the variation with temperature of the concentration of the activated complex at the reversible electrode, with no dependence upon the reference zero potential.
Introduction
xX
-x + b-B + {e:} 1-P Y V
x
The charge-transfer kinetics of electrode reactions have been interpreted most extensively by the theory of absolute reaction There are, however, some unsatisfactory aspects of the present rate equations: (1) the energy of activation depends upon a reference zero potential which is not accessible experimen tall^^-^; (2) the reaction rate may be expressed in terms of the concentration of the activated complex in its standard state, whereas the standard state is open to arbitrary selection6; (3) the activity coefficient of the activated complex is usually assumed to be unity, but it may have a profound influence on predicted rates and on allowable reaction mechanisms.’ The theory is modified below in an attempt to eliminate these drawbacks. A rigorous thermodynamic treatment is given for the activity of the activated complex. This requires definite values for the activity of the complex at the standard electrode and leads to new expressions for the concentration of the complex and reaction rates, in which all terms are independent of reference potential. The generalized electrode reaction utilized by Parsons3 will be treated with little change in notation. The over-a,ll electrode reaction is written
+ bB” + ne = WW 4-yY(bP-n)’y
(1) where either B and Y are both ionic, or only one is ionic (z = 0 or bx = n). The reaction may occur as a series of consecutive reaction steps. In the cathodic direction the component reactions are formulated as The Journal of Physical Chemistry
P
*Ao --f Q (rate determining, cathodic) (3)
+ ’Y
&+-W w, V
V
(4)
where v is the stoichiometric number. The total number, njv, of electrons is placed in braces to signify that the number of electrons used up in forming each of the intermediate states P, *&, and Q,is not fixed. The anodic reaction steps are the same, but in the reverse direction, with (3) rewritten
Q
*A,
--f
P (rate determining, anodic) ( 5 )
The cathodic activated complex *& has the same chemical constitution as the anodic, but its velocity along the reaction coordinate is in the opposite direction. At a given moment the concentration *co of the cathodic complex at the metal-solution interface will generally differ from *ca, the concentration of the (1) H. Eyring, S. Glasstone, and K. J. Laidler, J . Chem. Phys., 7 , 1053 (1939).
(2) S. Glasstone, H. Eyring, and K. J. Laidler, “Theory of Rate Processes,” McGraw-Hill Book Co., Inc., New York, N. Y., 1941. (3) R. Parsons, Trans. Faraday Soc., 47, 1332 (1951). (4) P. Delahay, “New Instrumental Methods in Electrochemistry,’’ Interscience Publishers, Inc., New York, N. Y., 1954,p. 42. (5) J. O’M.Bockris, Mod. Aspects Eleetro-Chem., 1, 180 (1954). (6) T.Hurlen, Electrochim. Acta, 7 , 653 (1962). (7) J. Horiuti and H. Sugawara, J . Res. Inst. Catalysis, Hokkaido Univ., 4, 1 (1956); J. Horiuti, ibid., 4, 55 (1956).
MODIFIEDGENERAL THEORY OF CHARGE-TRANSFER ELECTRODE KINETICS
anodic complex. The fundamental rate equations of the absolute reaction rate theory2may be written
kT rc = -xc h
*cc
kT ra = -xu h
*cB
Thermodynamics of the Activation Process The over-all cathodic activation process is represented by the equilibrium (omitting the electrons involved) V
+ b-B
*Ac
V
(7)
The first and second laws of thermodynamics applied to this process at constant T and P give *AUo = T*ASc - (P*AV, *u,),or -*AG, = u0 (8) where *AU,, *Ah‘,, ‘AV,, and *AG, are the changes in internal energy, entropy, volume, and Gibbs free energy, respectively, and *u, is the total electrical work done during cathodic activation
+
*Uc =
*
p
+
(9) The last term is a quantity not directly accessible experimentally. It can be expressed, following Horiuti U(P+*A)
and Polanyi,8 as a fraction 0 of the electrical work done in the rate-determining step P{(n/v)FA#I - P
U(P+*A) = PU(P-Q) =
where rc and ra are the cathodic and anodic rates and xo and xa are the transmission coefficients (assumed constant). Electrical work done during the formation of intermediate P from the cathodic reactants will be designated as p , and that done during the formation of Q from the products will be q. The quantities p and q will be restricted by the following conditions. (a) There may be portions of p and of q which depend on T and composition, hence which vary with the reversible potential, but which are independent of chargetransfer polarization r. (Note that Parsons* described these portions as independent of any change in singleelectrode potential.) (b) The remainder of p and q is directly proportional to the electrode potential, the proportionality constant being independent of T and composition. (c) The rate-determining step involves a finite quantity of electrical work; hence p - q is less than the total electrical work involved in the over-all reaction; either p or q or both may be zero. An inherent difference from Parsons’ treatment has been introduced to simplify discussion of the concepts being modified: the reaction sequence (2) to (4) includes only mechanisms for which the cathodic complex is formed from X and B in the ratio given by (l),thus excluding possible mechanisms in which the complex is formed from the same constituents in other proportions, or from B without X, etc.
x -X
413
+q)
(10) where A 4 is the total potential change across the electrode-solution reaction zone. With (9) and (lo), eq 8 becomes
-*AGO = P(n/v)FA#I
+ (1 - P)p -I- Pq
(11)
Here one introduces a fundamental assumption of electrode kinetic theory that the “transfer coefficient” p (0 < p < 1) is independent of T and P, of electrode potential, and of the concentration of any reactant or product of the electrode reaction. This constancy prevails for the range of conditions within which the reaction mechanism is invariant. The same assumption is applied to the transmission coefficients. The energy change (n/v)FAt$ is not experimentally accessible and not an absolute quantity; hence, the related free energy of activation, *AGO, is also not absolute. Since ions are involved, these quantities depend upon a reference level for ionic free energies. For consistency with tabulated thermodynamic data based upon the universally accepted rules for selection of standard states, these energy changes are based upon zero free energy of formation for hydrogen ion and upon the standard hydrogen electrode as the reference zero for single-electrode potentials. Therefore A+ in (11) may be replaced by the electrode potential E
+
+
- * A G ~ = P ( ~ ~ ) F (1 E - P)P Pq (12) for both A#I and & represent the measured voltage of an isothermal cell made by combining the test electrode and s.h.e., provided the measurement excludes any appreciable I R drop or liquid junction potential. With the boundary conditions assigned above to p and q, one may use Parsons’ equation (1
- PIP
+ Pq = Y ( d V ) F & + 6
(13) to define y and 6. The proportionality constant y is assumed to remain constant over the same range of conditions as 0, but y unlike p may be negative. The quantity 6 is independent of 17 but may vary with T and composition. Equation 12 may now be rewritten
- *AG
=
(P
+ Y)(W%FE + 6
(14) Let the concentrations of the electrode components be adjusted such that reaction 1 occurs with each inJ. Horiuti and M. Polanyi, Acto Physieochim. U R S S , 2, 505 (1935). (8)
Volume 70,Number 8 February 1966
414
SIDNEY BARNARTT
dependently variable reactant or product J in its standard state (activity U'J = 1). The standard state is conventionally the state of the pure substance at 1 atm for those reactants which may be present as solvent or pure substances in a separate phase. For dissolved solutes the standard state is given by aoJ = 1 under the condition of unit activity coefficient a t infinite dilution, i.e., U J -+ CJ as CJ + 0. If the reaction is carried out reversibly at a'~ = 1, we have the defined standard electrode at potential GO, and the standard free energy of activation becomes from (14) -*AG~,,
=
(P
+ T ) ( ~ E / ~ ) F +E O6,
(15)
where 6, is the value of 6 for the standard electrode. The activated complex, like any other dissolved solute, has its standard state defined by 'a', = 1 under the infinite dilution condition for solutes: *a, + *cc as *ec -+ 0. The free energies of activation in (14) and (15) are related by *AGO = *AGc,,
+ RT In [2&?5>""(
c$)""] (16)
where *ac,, is the activity of the cathodic activated complex at the standard electrode. Since each independently variable component J is at fixed (unit) activity at the standard electrode, the latter is completely defined, and therefore the equilibrium which exists between reactants and activated complex determines the chemical potential of the latter without further arbitrariness. It follows that the activity *ac,aof the complex at the standard electrode is a definite value established by the nature of the electrode system. It! is not equal to the standard state value, has a definite value for *aoc = 1. The fact that a given electrode system negates the validity of an arbitrary calculation of *a,,,, such as the one recently proposed,6 which does not take into consideration the detailed atomistic mechanism of the electrode reaction. The two conditions, thermodynamic reversibility and electrode potential E o , are necessary but not sufficient to establish the activity of the complex at the definite value *ao,,. The reversible electrode potential is given by = Eo
RT + nF -In-----
QX'UB~
aw"uy'
from which it is evident that one may obtain G, = Go at values ax # 1 and/or aB # 1 by adjusting the activities of the denominator substances to keep the activity product ratio unity. If the reversible potential G o is set up in this manner for aB # 1 (and/or ax Z l), The J O U T Mof ~ Physical Chemistry
the activity *ac which results from equilibrium 7 must differ from the value of *ac,, which develops at the standard electrode where U O X = aoB= 1. The latter condition is thus the third requirement for establishing *ac,a.
From eq 16 one obtains, after substituting the expressions given in (14) and (15) for *AGO and *AGO,, and unity for a0x and aoB
+
(18)
where E = F/RT. Then with & = E, 7, where 7 is the charge-transfer polarization and E, is given by (17), one obtains *ac/*ac,s
=
+
+
rI exPl-(P Y)(n/V)€? (6s where II denotes the activity product rI = ( a x z a B ~ ) ( ' - B - ' ) / Y ( a w ~ a y Y )
- 6)/RTl (P+')/V
(19) (20)
The thermodynamics of the anodic activation process may be treated in the same manner as above. Here the electrical work u(Q- SA) is the complementary fraction, 1 p, of the total for the rate-determining step. With this modification it is readily shown that the corresponding equations for anodic activation are *AG, = (1 - p - ~ ) ( ~ E / ~-) 6F E (21)
-
(22) *aa/*aa,s
=
l~ exp[(1
- P - r>(n/v>atl+ (6, - 6)/RTl
(23)
where the activity product II is identical with that for the cathodic activation (eq 20). Concentration of the Activated Complex The surface concentration of the activated complex will always be very small. This follows directly from (6) since k T / h is very large (6.2 X 10l2 at 300OK). For example, the value of * c corresponding to rapid discharge of a univalent ion at 1.0 A/cm2 is 1.0 X lo6 molecules/cm2 for x = 1. Since a uniform electrode surface will present roughly 1016 adsorption sites/cm2, this value of *c corresponds to a fractional surface coverage *e = Even with changes of three orders of magnitude in discharge rate, in transmission coefficient, or in available surfece sites, *e will remain very small.
MODIFIEDGENERAL THEORY OF CHARGE-TRANSFER ELECTRODE KINETICS
If the activated complex is the only species specifically adsorbed, the infinite dilution condition that *a + *c as *c + 0 permits replacement of the activty ratio in (18), (19), and (23) with the corresponding concentration ratio. This replacement will be valid also where other species are adsorbed, each to surface concentrations greater than *c, provided the total surface coverage by all remains small with respect to unity. This may be shown by use of the statistical t,hermodynamic treatment of a localized monolayer. Following Fowler and Guggenheim, one may express the activity ratio as
The concentration then *Cc,r/*Cc,s
415
bC,,at =
the reversible potential is
II exp[(bs - 6)/RTI
(28)
Similarly, for the anodic direction *ca/*c.,s =
II exp[(l
- P - r)(n/v)erl + (6, - 6)/RTl
(29)
and
Rate Equations (24) for the condition of no mutual interactions between adsorbed molecules. Here &e, is the total fractional coverage of electrode surface sites by all specifically adsorbed species, both charged and uncharged, for any given set of solution conditions and potential E ; ZtOi,, is the corresponding quantity at the standard electrode. For electrode systems which maintain ZfOf (n/v>s+ (6, - 6)/RTI
(27)
The rate equations are obtained by substituting * c from (27) and (29) into (6)
rc = ( k : T / h ) x , + c , , , ~exp[-(p ~
+ r)(n/v)elt +
- S)/RT] = & / ( n / v ) F (31) (kT/h)xa*ca,sII exp[(l - P - r)(n/v)elt+ (6s - b)/RT] = i a / ( n / v ) F (32) (6,
ra
=
where ic and i, are the rates expressed as current densities. The rate in either direction is proportional to II; hence, both rates have the same dependence on the activity of any substance taking part in reaction 1 whether reactant or product. In the present formulation of the reaction rate, the activity coefficient of the activated complex enters as the ratio *fs/*f, which is unity for all of the reaction categories treated. I n the previous f o r m u l a t i ~ n ~ ~ ~ ~ ~ this term appeared as a single activity coefficient, and, hence, within category C-3 (activity coefficient >1) it would affect the reaction rate. The use of eq 24 to evaluate *a/*a, implies that the adsorbed species on the electrode surface obey the Langmuir isotherm. If another isotherm were obeyed instead (Parsonslo describes 10 common isotherms) , the activity coefficient term could not generally be neglected. Here, it would be sufficient to derive *a/*a,and thence *fs/*f for the particular isotherm and insert the latter as a multiplying factor in the expressions for concentration of the complex (eq 27, 29) and reaction rate (eq 31,32). In contrast with previous formulations, the present one clearly demonstrates the independence of reaction rate on the reference zero of potential. The reaction rate is proportional to *C (eq 6) or to *a, and the (9) R.Fowler and E. A. Guggenheim, “Statistical Thermodynamics,” Cambridge University Press, London, 1939, p 421. (10) R. Parsons, “Proceedings of the Fourth Soviet Conference on Electrochemistry, 1956,” Consultants Bureau, New York, N. Y., 1961, p 18.
Volume 70, Number 8 February 1966
SIDNEY BARNARTT
416
latter was shown (eq 18) to be determined by the difference between two values of the single electrode potential, as well as by *ae which has a definite value fixed by the nature of the electrode reaction. Therefore the rate equations (31) and (32), which derive directly from (18), arc independent of the selected reference potential level. The activation energy for the reaction, obtained from the variation of the rate constant with T (see below), must also be independent of reference potential level. The effect of 6 on reaction rate is now different from that derived by parson^.^ The latter treatment yielded rates proportional to exp[--G/RT], so that 6 affected the rate as long as its value was an appreciable fraction of RT. It is clear from the present formulation that 6 can be large and still have little or no effect on the reaction if variations in T or composition produce relatively small changes in 6.
Exchange Currents At the reversible potential i, = ia = io, the exchange current density; hence
-xaTca,,ii exp
h
LRT 1
If the electrode is in its standard state, II
1, 6 = 6,, and the rate in either direction is given by the standard exchange current density io,8
&,,/(n/v)F= (kT/h)xc*Cc,,
=
=
these two concentrations is not changed by change in T o r composition. The concentration of the complex a t the standard electrode, ‘c,,, or *Ca,s, may be considered the fundamental parameter of this modified general theory. Therefore a successful atomistic mechanism of a specific electrode reaction should be able to predict *cs and its temperature variation. The standard rate constant io,,/(n/v)F, which is proportional to ‘c, (eq 34), may be identified with the quantity k s , h introduced in 1954 by Delahay.” The rate equations may be written in terms of the standard exchange current density
ic = io,,II exp[(6,
- 6)/RT]
X
expI - (P
ia = io,,II exp[(6,
+ r)(4~) €71 (39)
- 6 ) / R T ]X exp[(l - B
- r)(n/v>vl
(40)
For 7 negative, the net (positive) cathodic current density is i = ic- ia,or
For 7 positive, the net (positive) anodic current density is i = ia - i,, or
x
i = io,,nex”[%]
(kT/h)xa*c8,, (34)
whence xc*~c,s
=
(35)
xa*ca,s
Equation 33 may now be rewritten io =
io,s
IIexp[-] 6, - 6 RT
=
These equations are readily simplified for the following two groups of mechanisms. (A) All of the electrical work preceding and following the rate-determining step is proportional to the single electrode potential; 6 = 0 and (41) reduces to
i By combining (36) with (30) one obtains
i,,/io,,
=
+cc,r/*Cc.s
=
*Ca,r/*ca,S
(37)
and this with (35) gives
.
-‘cc,r/
*
ca,r
*cc,r/*Ca,s
= x8/xC
- (j? + 7)”.7]{ V
1 - exp( :e?)}
(41A)
(B) Xo electrical work is involved in the steps preceding and following t’he rate-determining step; y = 6 = 0 and
(38)
The last result expresses two conclusions concerning the electrode a t any reversible potential. (1) The concentration of the cathodic activated complex will be equal to that of the anodic complex only when the transmission coefficients are equal. (2) The ratio Of The Journal of Physical Chemistry
= io,JI exp[
i
= i 0 J I ’ exp[
--~Lv]{ 1
- exp(})?E:
where
II’ =
b (1-8)/v (UXZUB )
(11) P. Delahay, ref 4,
35,
(a~vwaYy) @’”
(41B)
MODIFIEDGENERAL THEORY OF CHARGE-TRANSFER ELECTRODE KINETICS
Reaction Order The exchange current density given by (36) is the basic experimental quantity for determining the order of the reaction, WJ, with respect to any reactant J t,aking part in the reduction or the reverse (oxidation) rcxtion. Thus, for ion B uB=(-)
b In io b In U B
417
where WB is the reaction order as determined potentiostatically. Similarly for constant anodic current in the Tafel region, the effect of any reactant, such as Y, is
= T,aj#~
V
The anodic and cathodic reaction rates exhibit exactly the same dependence
Activation Energy The exchange current density is also the basic experimental quantity for determination of the characteristic activation energy
(Egi) =
II
Here it must be understood that values of io and i,, utilized in (44) are the total cathodic or anodic current densities, which for fast reactions at small q are not the experimentally observed net current densities. If activity coefficients of B are unknown, one may determine io in solutions containing excess inert electrolyte in which the activity coefficient remains constant; then the concentration CB may be substituted for U B in (43) without error. The reaction order with respect to reactant Y of the oxidation reaction is from (36)
The reaction order with respect to X or W is obtained similarly from io. If one of the substances J has little effect on 6 the reaction order reduces to the first term of the appropriate equation. The alternative definiq), tion of reaction order in terms of constant (&, e.g., by VetterlX2leads to different values depending on whether the order is obtained from i, or from i,,, an unnecessary complication. The experimental procedure for constant (&, q) is admittedly more direct since the reaction rate can be followed potentiostatically while incremental concentration changes are made to the solution. Only a minor change in procedure, however, may be required to obtain the measurements at constant 7, namely, a corresponding change in potentiostat setting of A&r applied as each concentration change is made. The reaction order can also be determined experimentally by galvanostatic measurements. It is readily shown from (39) that, for an applied constant cathodic current in the Tafel region, the effect of varying the activity of any reactant, for example, ion B, is given by
+
+
E*,, RT2
With (33) and (30) one obtains
(49) where * c ~ and , ~ *cC,. may be replaced by *c,,,, and *calS, respectively, without effect. This expression is quite different from that given by BockrisI3 for the “virtual heat of activation,” and unlike the latter it yields values independent of the reference potential level. Equation 49 shows that the experimental energy of activation may vary with solution composition. For the electrode in its standard state, 6 = 6,, and one obtains the standard activation energy
E
+
)
d In *ccVs
~= RT ~ ~ R T, ~ (~ d T
(50)
Other Mechanisms The family of mechanisms to which the above development applies was restricted by specifying that the activated complex contain reactants X and B in the ration x :b and also that p and q comprise potentialdependent and composition-dependent portions having properties defined by the quantities y and 6. In this section these restrictions will be removed to permit analysis of other classes of mechanisms. The over-all electrode reaction per mole of activated complex may be written (12) K. J;, Vetter, “Transactions of the Symposium on Electrode Processes, John Wiley and Sons, Inc., New York, N. Y . ,1961,p 47. (13) J. O’M. Bockris, Mod. Aspects Eleetro-Chem., 1, 197 (1954).
Volume 70, Number 2 February 1966
SIDNEYBARNARTT
418
We will now permit the cathodic complex to be formed by reaction of z' 5 z/v molecules of X, b' 5 b / v molecules of B, and n' 5 n/v electrons
zfX
+ b f B + n'e
*Ac
--f
Q
-
-
(62)
(52)
The electJrical work done in the rate-determining step, P ic Q, may be either finite (case A) or zero (case B). I n either case the free energy of activation becomes, for the cathodic process
+ RT In [
(61)
For the reverse reaction
'("
*AGc = *M,,,
P(qsRT - )' exp[ - f e 7 ]
(54)
RT
exp[ (1
- P)5~7]
At the reversible potential p = p,, q = qr, and the exchange current density given by (61) or (62) is
(63) The rate equations, rewritten in terms of io, become
For the anodic activation equilibrium "-W+-YV
V
*A,,
(4
+ - xf>X +
+ RT In X
*AG, =
Case A . The electrical work done in the ratedetermining step is finite. The free energy change for the equilibrium in (52) is still given by (12) -*AG,
+ (1 - P)P + Pq = P(n/v)FE" + (1 - P)ps f PqB
=
- *AG,,,
P(n/v)FE
(57)
(58)
where p s and qs are the values of p and q when the electrode is in its standard state. With (57), (58), and (17) one obtains from (54)
-+ao - - IT" exp (1 *ac,s
where
- P>(Ps - PI RT
I T f p = axz' - (Bx/p)aBb'-
X
(Bb/u)aw@w/vayBv/v
io = io.811ffexp[(bz - pn)(e/v)(A (60)
After equating this activity ratio to the concentration ratio and combining with (6) and (34), one obtains the cathodic rate equation The Journal
of
Physical Chemistry
A specific mechanism will dictate expressions for the exponentials containing p or q in terms of experimentally accessible parameters. Case A-1. Modified Frumlcin Equation. As an illustration we will apply these rate equations to the diffuse double layer effect treated by Frumkin,14 with discharge of ion Bz+ as the ratedetermining step. Here electrical work p results only from the movement of b/v ions B through the diffuse double layer potential +, so that p = bzF+/v, where $ will generally vary with the polarization 9 as well as with T and composition.'6 Similarly, q results only from the movement of y/v ions Y through the same potential difference; hence q = (bx - n)F+/v. Substitution for p and q in (61) and (62) yields
- +)I
X
exp [ -P ( n / v > €71 (66) (14) A. N.Frumkin, 2.Physik. Chem., A164, 121 (1933). (15) R. Parsons, Advan. Electrochem. Electrochem. Eng., 1, 1 (1961).
MODIFIEDGENERALTHEORY OF CHARGE-TRANSFER ELECTRODE KINETICS
ia
= io,.TI’’
exp[(bz
- P n ) ( e / ~ ) ( +-~+)I X exp[(l - P)(n/v)elll
(67)
+
If varies appreciably with 9 , the electrode reaction rates will not obey the Tafel equation. These rate equations are not identical with the generalized Frumkin equations given by Parsons,16 the principal in Parsons’ modification being that the quantity equations is now replaced by (+E - +). Thus, for a range of solution composition over which - + remains small, even though + itself may be large, the diffuse double layer will have negligible effect on reaction rate. At the reversible potential, = +r and the exchange current density is
+
+.
419
- *AG,,, = pa. Substitution of these quantities into (54) yields *a,/
*a,,.
=
Pa
X’
-P
ax aBb’exp- RT
(73)
which leads to the cathodic rate equation
i,
= io,.aX2‘aBb’exp-
Pa - P RT
(74)
Similarly for the anodic reaction - *AGa = q and -*AGa,, = q.; these values substituted into (56) yield
+
io
=
io,.n” exp[(bz - P n > ( ~ / v ) ( + ~+r)l
(68)
I n terms of io the cathodic current density becomes
io = io exp[(bz - Pn)(E/v)(+r
- $11
X
exp [ -P(n/v> €71 I
(69)
and an analogous expression may be written fori,. The reaction order with respect to a given reactant may be obtained from (68). Thus, for ion B
I n this case also a specific mechanism will determine experimentally accessible expressions for the exponential term in each of the rate equations. These terms may be converted immediately if either p = 0 or q = 0. Thus, for p = 0 all of the electrical work of the reaction is given by p = ( n / v ) F & , so that (74) and (75) simplify to 2’- ( d v ) a B b ’ - ( b / v ) a W w / v a y u / v = io i a = &,.ax (76) i, = io exp[-neqlv]
(70) The experimental energy of activation, defined by (48), is also obtained from (68)
Electrical work is done only during the cathodic activation; hence, only the cathodic current density depends upon the charge-transfer polarization. The reaction order, defined as in (43), for each participating substance is given by the exponent of the corresponding activity in (76). Similarly for p = 0 the rate equations become
i, = iO,saXx’aBb’ = io
and is seen to vary with +r, i.e., with the particular solution composition selected for study. Case B. The rate-determining step involves no electrical work; hence p
-q
=
(n/v)F&
(72)
This condition simplifies eq 8 to -*AGO = p ; also
(77)
(78)
ia = io exp(nq/v) (79) and only the anodic current density is polarization dependent. Here the reaction order is simply x’ with respect to X, b’ with respect to B, and zero with respect to W or Y. Acknowledgments. The author gratefully acknowledges fruitful discussions with Drs. L. S. Darken, R. P. Frankenthal, and R. A. Oriani of this laboratory.
Volume 70,Number 2 February 1966
