THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in
U. 8.Patent Office)
VOLUME63
(0Copyright, 1959, by the American Chemical Society)
NUMBER11
NOVEMBER 18, 1959
KINETICS OF THE ELECTRODE PROCESSES INVOLVING MORE THAN ONE STEP BYB. LOVREEER~ John Harrison Laboratory of Chemistry, University of Pennsylvania, Penna. Received October 9, 1968
A method is developed for kinetics of electrode processes proceding in more than one step and a theoretical expression is found for the slope of the Tafel line for caBes in which one Step is slow and rate determining. Usin4 the experimental slope and this equation, i t is possible to determine both the a-value and the step which is rate determining. As a further consequence of such treatment, a corrected theoretical interpretation of the experimental ( 7 i )curve is presented. Th: approach of this method is different from those published earlier by otherseSaand more general than a similar one by Vetter.
-
There is evidence that many electrode processes proceed in more than one step! Thus the over-all reaction of an electrode process as represented by eq. 1 0
+ zeo- If R
(1)
(where 0 is oxidized form, R reduced form and z the number of electrons used in the reaction ac-
cording to eq. 1) may involve several steps 0
+ zleo- I _ P
+ zzeo- If Q Q + zaeo- J _ R
P
(24 (2b) (24
Every step represents an oxidation-reduction system, except in the case of electrochemical deposition or dissolution of a metal, where the last eq. 2c represents an electrochemical system Me/MeZ*+, or with the symbols from eq. 2 R/uQ, where UQ is the activity of the respective ion. If so, one can assume also that a t the reversible potential Vre, for the over-all reaction 1, when no net current flows, electrochemical reactions (eq. 2a-2c) proceed simultaneously and the reversible potential for the over-all reaction 1 is under the given conditions the reversible potential for each “reaction step” (2a-2c). The rate of each “reaction step” a t the reversible potential (and also (1) University of Zagreb, Zagreb, Yugoslavia. (2) R. Parsons, Trans. Faraday S O C .47, , 1332 (1951). (3) J. O’M.Bockris, J . Chem. Plays., 24, 817 (1950). (4) K.J. Vetter, Z. Naturforsch., 7a, 328 (1952); 8&, 823 (1953). (5) E.@., W. Kangro and F r . Weingartner, 2. Elektrochem. Eer. Bunsenges., 68, 505 (1954); 69, 137 (1955).
during the polarization, after a steady state is reached) must be equal.6 If a net cathodic or anodic current flows, an overpotential can be expected. Under the assumption that one of the steps is slow, i.e., rate determining, the following changes will take place at the electrode. For increase of the potential of the electrode by 0.059 v. at 25” (or 2.303RT/F), if every step involves 1 electron, a change in the ratio of activities, for every step which can be considered as reversible, will be one power of ten. If the step Q 4- 2aeo- a R is the electrochemical deposition or dissolution of a metal, the activity UQ will also change by one power of ten. This does not apply for the slow, irreversible step, but a method will be shown later by which the change of activities of the ions involved in the ratedetermining step can be evaluated. If the activity of the starting ion a0 in cathodic polarization is assumed constant,? the activities of other ions, preceding the rate-determining step, will change in the manner given for the change of potential, -0.059 v. at 25’ or (-2.303RTIF). For the first two steps, if they are reversible a ’ p = lOap a’Q = 10%~
(3) (34
( 6 ) The differences in diffusion rates may cause some differences in rates of electrochemical reactions during polarization t o maintain the steady-state concentrations. (7) This is the usual assumption if the bulk aoncentration of the respective ion is high and diffusion can replace discharged ions sufficiently fa&, so t h a t no appreciable concentration overpotential develops. Also a high aoncentration of a foreign electrolyte is assunied to eliminate the potential.
1795
r
B. LOVREEEK
1796
(a’ is the activity after the change of the potential and a is the activity before the change of the potential). This will give the new activity ratios corresponding to the change of the potential
Vol. 63
or log a’fl** =
- ‘IFAv - (’** 2.303RT
+ log an*,
(13a)
(the numbering of the steps begins from 0). If z is not equal to 1 in all steps, appropriate and changes should be made in equations 3 to 13a. For metal electrodeposition and dissolution the same equations can be applied if R is replaced by The change in activity ratio for each reversible Me. The above considerations can now be applied to step is one power of ten. But the absolute change in activity of an ion depends on the step in which the usual equations for electrode kinetics.* For a cathodic rate-determining step, if x = 1, and if it is involved. E.g. anodic current can be neglected (in presence of U’P = lOap (note that P will be reduced in the second step) excess of a foreign electrolyte) U‘Q - 1O2aQ (note that Q will be reduced in the third step) (4)
Fk
i-
or in general u’U
(5)
= 10m-lanc
(nc is the number of the step in which the respective ion will be reduced.) If n, step is rate determining the activity of the ion reduced in this slow step (a’n*,) is given by eq. 6 = 10n*c-1 an*c
a‘Bo 1O-(Ti0-1) ane a’ = activity after the change of the potential E
(7)
a
= starting activity of the Bame ion Z = no. of the step (numbering starts from R)
- 10-(7i*e-~)
i- =
(8)
- 1) + log ano
(9)
for every change of potential of -2.303RTIF volts, or for any change of potential (AV) log a’& =
- 1)FAV - (%2.303RT
also from eq. 7 log a’n, =
(‘0
+
5
(14) (14a)
K-an*ce-(l-u)FV/RT
From eq. 11 it follows that A log an*, =
- (n*c -
AV
1)F
2.303RT
(16)
i.e., the change of log an*, with V is a linear function of n *o and Combining eq. 16a and 15
a*, is the number of the rate-determining step (numbering starts from R). From eq. 5 it follows that log a’n, = (no
e-(l-a)FV/RT
(k- and K- are constants) and
a
And for the ion produced in the rate-determining SkP a‘a*,
- an*, e-AU-/RT
or
(6)
The activity of any ion, involved in reduction after the rate-determining step, can be calculated in a similar way, starting from the constant activity U R
I
P c
log- i-= bV
(n*o - l ) F 2.303RT
(1
- a)F
2.303RT
(17)
or
For the anodic rate-determining step, if I = 1, when cathodic current can be neglected, analogous equations can be derived
log
- ‘IFAV + log ano
2.303RT
or (loa)
or for the ions involved in the slow step and I n anodic polarization the same reasoning can be applied so that or
(the numbering of the steps begins from R) and
If x is not equal to 1 in the ratedetermining step, appropriate changes in equations 14 to 18a should be made. In the above expressions i- and i+ should be taken as that part of the whole current which is used in the rate-determining step. In a “many step” electrochemical reaction, after a steady state is reached, the part of the current used in each step is proportional to the number of electrons involved in the respective steps. But for the determination of the slope log i- (or log i+) versus V the usual log i-V representation (i is the total measured current) can be applied, since the calculation and graphical representation of the true (8) J. A . V . Butler, Trans. Faraday Soc., 19, 729 (1924); 19, 734 (1924); T. Erdey-Grua and M. Volmer, Z . physik. Chem., l5OA, 203 (1930).
c*
Nov., 1959
‘[step-current” would displace the curve in a parallel direction, but would not affect the slope. From eq. 15 to 18a one can see that on the basis of experimental relation log i-V the correct value of CY can be calculated from an expression which differs from the usual one9 by a correction which takes into account changes in the activity of the ion which is to be reduced or oxidized in the ratedetermining step. If the first step is rate determining the correction equals zero. Using this method the original physical meaning of CY is preserved. Further this method offers the possibility to get a fair estimation on which step is rate determining in a “many step” electrode process. This is so because the value of n* can only be a small integer and 0 < a < 1, so that in equations 17a and 18a the term in brackets, which can be evaluated from experiment, indicates the value of n* and a, e.g., if the term in brackets is less than 1, n* = 1; if the value is between 1 and 2, n* = 2, etc. With certain precautions also the usual equations
log io, log i. Fig. 1.
log io2
The term on the left-hand side of eq. 20a represents the shift of the reversible potential (AVrevc)for a change of polarization potential (AV)
and
AVrevc
can be applied, if care is taken that each term corresponds to its original definition: (a) One must apply them for the electrochemical process of the rate-determining step only and consequently i- or i+ can represent only this part of the current which is used in this step. (b) The activation overpotential is no longer simply the difference in potential at a certain current and in beginning of the measurement before polarization started. Because of the change of activities with polarization (eq. 10-13a) and in accordance with the essence of the definition of activation overpotential, qa should represent the differences between the potential during polarization and that without net current flow, but in solutions of the same activity of corresponding ions. The starting reversible potential should therefore be corrected for every potential obtained during polarization. This correction can be made using equations 11, l l a and 12a, 13a. From the first two, for cathodic polarization, it follows that an*, log a‘n*o - - log - = a ’I3 *a an*,
1797
KINETICS OF ELECTRODE PROCESSES WITH MORE THAN ONE STEP
FAV - 2.303RT ~
+ E*,
- 2)
-
AV
(%*a
+ 6*o
- 2)
(21)
For anodic polarization analogous calculations (from eq. 12a and 13a )give AVreva =
AV(n*,
+ Z*,
- 2)
(22)
Using equations 21 and 22 the true 7. can be calculated from measured values. Finally the io value which is by definition given by the equation io
=
zFk-
a0
exp
[- AG-
+ RT
=
should also be calculated taking into account the change of both activity and Vre,. This can be done graphically in the usual way from experimental log i-V curves, but by extrapolating the straight portion of the curve for every measured potential to a new value of reversible potential, as schematically represented in Fig. 1. With these precautions the usual kinetic equation for electrode processes can be applied also in a “many step” electrochemical reduction or oxidation. The author wishes to thank Prof. J. 0. M. Rockris for helpful discussion of this work. (9) Derived from eq. 14 or its equivalent for anodic reaction, but without taking into account the changes in the activity of the ion which is t o be reduced or oxidized in the rate-determining step.
