J. Phys. Chem. 1983, 87, 2492-2502
2492
A
I
order of magnitude as those obtained for weak electrolytes in low-dielectric media: e.g., KT 102-103; K , 1 102.% The comparisons rely to a large extent on u trasonic measurements, but other techniques-5"-jump, for exampleyield similar estimates. K, tends to vary a good deal more than KB or KT, but the value of 5 X lo5 does not appear unreasonable. Lithium salts in various low-susceptibility solvents give K, i= 107-109 M-l.,* The rates calculated for dimer formation from monomer and trimer deserve comment; the main difference in the rates obtained for equilibrium C, as opposed to the rates calculated for equilibrium IV, is that, in the all-over kinetics for equilibrium C, we get linear behavior for 1/r with c, whereas in equilibrium IV we get (at best) a square-root relation. It is hard to propose a consistent mechanism leading to this concentration dependence without invoking some form of higher aggregation. The questions of the role of hydrogen-bond formation and tautomerization of the azo and amino nitrogens remain for further study. We have really sidestepped these issues by picturing a dimer of the form (BHA),; the internal structure of such a dimer may involve hydrogen bonding, azo/amino tautomers, or both.
7 -
.>I
L
50
io 0
WeY
is 0 concemtret.on
do0
azo
( / H )
Figure 4. Least-squares flt for 117 vs. concentration of M Y in W A C .
106 M-l. Analysis of the conductance data by the Wooster method gives KT = 138 M-l. Figure 4 shows the E-jump 1 / vs. ~ concentration data for MeY. The computed slope and intercept are slope = 2.13 X lo9 M-l s-l, intercept = 2.80 X lo5 s-l, so that kf = ( ~ l o p e ) / 2 ( K ~ / K ~=) '12.8 / ~ X 1O'O (M s)-l, in reasonable agreement with the computed diffusion-controlled rate of 8.2 X 1O'O (M s)-l. The ratio kf/kd = K, = 4.57 X lo5 s-l, and from the other three equilibrium constants
Acknowledgment. This research was partially supported by DOE contract no. OE-AC02-78ER14945, and by the Rutgers Univesity Research Council. The work comprised part of R.J.R.'s Ph.D thesis, submitted May 1982. Registry No. Methyl yellow, 60-11-7; acetic acid, 64-19-7.
KB = KT/Kx/Kp = 32 M-' Discussion The scheme for aggregate formation of MeY in HOAc represented by mechanisms A-D proved to be consistent with the experimental kinetic results and the conductance data. The derived values for KT and KQare of the same
(23) Saar, D. D. Doctoral Dissertation, Polytechnic Institute of New York, New York, NY, 1980. (24) Macanka, W. Doctoral Dissertation, Rutgers University, New Brunswick, NJ, 1981.
Electrochemical Reduction of an Isomeric Pair When the Products Interconvert Alan M. Bond' and KeRh B. Oldham'+ Division of Chemical and Physical Sciences, Deakin University, Wawn Ponds 3217, Vicforia, Australia (Received: November 15, 1982)
There is a commonly held view that structural change accompanying an electron-transferreaction leads to the observation of sub-Nernstian behavior. This may be reasonable if the straightforward mechanism ne + 01 s R2 is operative. However, if part of the "square" scheme
= RI
ne
t 01
ne
t 02 a R 2
It
If
applies, this conclusion is shown not to be generally valid. The converse thesis that Nernstian behavior implies structural integrity is also not generally correct. Reactions that are part of the square mechanism may cause a reversible electron transfer to appear irreversible by decreasing the slope of the reduction wave. But our treatment shows that they may equally well increase the slope or leave it unchanged. Under other circumstances the reduction wave may split into two, or become preceded by a gratuitous peak. The present study convinces us that drawing mechanistic conclusions from voltammetric data alone is fraught with danger. 1. Introduction
Considerable controveq has arisen in recent years Over claims of slow electron transfer being due to conformational changes.l It is often assumed that structural inferences can be drawn from the observation of reversibility, Permanent address: Trent University, Peterborough, Canada. 0022-3654/83/2087-2492$01.50/0
or the lack of it, in an electrochemical process.2 Thus if a species 01 is observed to undergo a reversible electroreduction, it is inferred that the product must be species Organometal. (1) C. P. Carey, Chem., L. 15b, D. Albin, c37(1978). M. C. Saeman, and D. H. Evans, J. (2) A. J. Bard and L. R. Faulkner, 'Electrochemical Methods", Wiley, New York, 1980, p 114.
0 1983 American Chemical Society
Electrochemical Reduction of an Isomeric Pair
R1, isostructural with 01, rather than some isomer R 2 whose structure resembles the corresponding oxidized species 02. The argument that leads to this inference is that an interstructural conversion 0 1 0 2 or R1- R 2 would induce irreversibility into the overall reaction and destroy Nernstian behavior. For example, Kojima and Bard3 reported very large heterogeneous charge-transfer rate constants for reductions of some aromatic hydrocarbons in which structural conversions are precluded by stereochemical considerations. These authors attribute the extremely fast charge transfers to the absence of structural rearrangement. Similarly, Tulyathan and Geiger4ascribed the fast electroreduction of (cyc1ooctatetraene)iron tricarbonyl to the preservation of the structure of the organic ligand. They contrasted this behavior with the slow reduction of cyclooctatetraeneitself, in which the tub-shaped neutral molecule is converted to a planar anion."1° Other examples in which slow electron transfer has been correlated with conformational or structural changes include the redox reactions of substituted hydrazines,l' the hexaosmium octadecacarbonyl cluster,12 and various cobalt(II1) c~mplexes.'~J~ On the other hand, and in contrast to the above examples, there are some well-documented instances in which the electron transfer appears rapid, despite gross structural changes. The reduction of thallium(1) to its amalgam at a mercury electrode is a classic case in point: the process is Nernstian15 despite massive structural differences between the initial and final states. Similarly the anodic oxidation of ci~-Mo(CO),(dpe)~ [where dpe = 1,2-bis(diphenylphosphinoethane)] leads to isolation of trans-Mo(CO),(dpe),+ but occurs via Nernstian charge transfer.I6 The field of electrochemically induced provides other counterexamples to the thesis of a correlation between the rapidity of electron transfer and structural integrity.
-
(3)H. Kojima and A. J. Bard, J. Am. Chem. SOC.,97,6317 (1975). (4)B. Tulyathan and W. E. Geiger, J. Electroanal. Chem., 109,325 (1980). (5)R. A. Allendoerfer and P. H. Rieger, J. Am. Chem. SOC.,87,2336 (1965). (6)B. J. Huebert and D. E. Smith, J. Electroanal. Chem., 31, 333 (1971). (7)D.R. Thielen and L. B. Anderson, J. Am. Chem. SOC., 94,2521 (1972). (8)A. J. Fry, C. S. Hutchins, and L. L. Chung, J. Am. Chem. SOC.,97, 591 (1975). (9)H.Kojima, A. J. Bard, H. N. C. Wong, and F. Sondheimer, J.Am. Chem. SOC.,98,5560 (1976). (10)L. A. Paquette, J. M. Gardlik, L. K. Johnson, and K. J. McCulloch, J. Am. Chem. SOC.,102,5026 (1980). (11)S. F. Nelson, E. L. Clennan, and D. H. Evans, J. Am. Chem. SOC., 100,4012 (1978),and references cited therein. (12)B. Tulyathan and W. E. Geiger, private communication. (13)J. F. Endicott, R. R. Schroeder, D. H. Chichester, and D. R. Ferrier, J.Phys. Chem., 77,2579 (1973). (14)M. D. Glick, W. G. Schmonsees, and J. F. Endicott, J.Am. Chem. SOC.,96,5661 (1974). (15)N.Tanaka and R. Tamamushi, Electrochim. Acta, 9,963(1964). (16) R. L. Wimmer, M. R. Snow, and A. M. Bond, Inorg. Chem., 13, 1617 (1974). (17)R. D.Rieke, H. Kojima, and K. Ofele, J.Am. Chem. SOC.,98,6735 (1976). (18)J. Moraczewski and W. E. Geiger, J.Am. Chem. SOC.,103,4779 (1981). (19)A. M. Bond, D. J. Darensbourg, E. Mocellin, and B. J. Stewart, J. Am. Chem. SOC.,103,6827 (1981),and references cited therein. (20)K. M. Kadish, K. Das, D. Schaeper, C. L. Merrill, B. R. Welch, and L. J. Wilson, Inorg. Chem., 19,2816 (1980).
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983 2493
In discussing the reduction21of a compound 0 1 to the isomer R 2 of its isostructural reductant R1, one must recognize that there are at least two pathways alternative to the direct reduction
0 1 + ne
F!
R2
(1.1)
The isomerization may precede 01
It 02
(1.2) ne
R2
or succeed 01
ne
R1
If
(1.3)
R2
the electron transfer. One of the most general indirect mechanisms would incorporate all the processes shown in the following "square" scheme: 01
It 02
ne
ne
RI
It
(1.4)
R2
In some experimental s t ~ d i e ~ , ~ ~evidence J ~ J ~ has p ~been ~,~~ found for the participation of three or all four of the species 01,02,R1, and R2, but 0 2 and/or R1 may, of course, play roles in reductions in which they have not been detected. The reactions shown in (1.4) do not exhaust the possibilities. Second-order reactions such as
02 + R1
F!
01 +R2
(1.5)
may participate and yet other processes may be important in particular cases. The electrochemical consequences of reaction scheme 1.1 are ~ e l l - k n o w n . ~Under ~ potentiostatic conditions (polarography, chronoamperometry, normal pulse polarography, and similar techniques) a single wave is generated, the slope of which provides information about the reversibility of the electron transfer, as does the time dependence of the current. The results of nonpotentiostatic experiments (linear sweep voltammetry, cyclic voltammetry, differential pulse polarography, etc.) can be analyzed to provide the same information. The so-called CE and EC mechanisms, schemes 1.2 and 1.3,have also received considerable theoretical attention25 but the complete square scheme 1.4 is unlikely to be mathematically tractable. Indeed, a full solution to the square scheme would be of little practical utility anyway, inasmuch as it would contain seven a-priori unknown kinetic parameters. Nevertheless, any critical evaluation of correlations between structural changes and electrochemical behavior must address mechanisms more elaborate than the direct scheme 1.1. (21)Most of our discussion deals with reductions, but similar argumenta hold for electrochemical oxidations. (22)G. Pilloni, G. Zotti, C. Corvaja, and M. Martelli, J.Electroanal. Chem., 91,358 (1978). (23)R. A. Rader and D. R. McMillin, Inorg. Chem., 18,546 (1979). (24)See, for example, ref 2, p 158. (25)See, for example, D. D. Macdonald, 'Transient Techniques in Electrochemistry", Plenum Press, New York, 1977.
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Bond and OMham
The Journal of phvsical Chemlstty, Vol. 87, No. 14, 1983
For potentiostatic electrochemical conditions we have found analytical solutions for the four schemes ne
01
e
RI
It
ne
ne
01
e
02
==
01
ne, R t
RI
ne
ti
ne
R2
it
02
e R2
01
R1
11
11
02
$k R 2
(1.9)
each of which incorporates a realistic subset of the processes that contribute to the complete square scheme. [In diagramming these schemes we use to indicate an equilibrium and e to indicate opposed reactions each with a finite rate constant.] The solutions for schemes 1.7 and 1.8 will be reported in future publications; the derivation of the solution for scheme 1.6 is presented in the next section. Scheme 1.9, in which all steps are at equilibrium, inevitably yields a single Nernstian wave irrespective of the details of each process. 2. General Theory
Let 01 and 0 2 be a pair of structurally dissimilar isomers that are reversibly reduced to the isomeric products R1 and R2. We denote the standard (more strictly the formal) potential of the Oj/Rj couple by E O , where j = 1 or 2. The reactants do not isomerize directly, but R j converts to RJ (where J = 3 - j ) with a homogeneous first-order rate constant kj. The reaction scheme is thus 01 t
ne
02 t ne
*
R1
R2
where n is the number of electrons transferred in the reduction. The cell solution contains, in addition to excess supporting electrolyte, bulk concentrations cbl and cb2of 01 and 0 2 , respectively, but is initially devoid of R1 and R2. The chemically inert electrode has an area A and its potential is stepped, at time t = 0, from a value sufficiently positive that the current i is negligible, to a more negative value E. We seek to determine how subsequently i depends on E, t, and the other parameters of the system: the concentrations cbj, the rate constants k,, and the various diffusion coefficients. The usual voltammetric assumption of transport solely by semiinfinite planar diffusion will be made. I t will be assumed that R1 and R2 share a common value D of diffusion coefficient, but the diffusion coefficients of 0 1 and 0 2 will be treated as distinct and equal to D1and D2, respectively. Fick’s second law
Flguro 1. Plots vs. potentlal of the functlons gl, g2,g , and y. In drawing all the curves the half-wave potential separation Eh2- Ehl was taken as 8RTInF. Additionally a k l l k p ratio of 19.215 was assumed in constructing the g and y curves.
requires no modification for species 0 1 and 0 2 , but terms must be appended
to account for the kinetics in the cases of species R1 and R2. In these equations x represents distance measured normally from the electrode surface and each c denotes the concentration of the subscripted species at distance x and time t. The Faradaic current resulting from the reduction of O j is given by Fick’s first and Faraday’s laws as (2.4)
where F is Faraday’s contant and a superscript s is used to denote conditions at the x = 0 electrode surface. Because each reduction is reversible, a Nernst equation links the surface concentrations Po; and cap, of the members of each redox pair. This Nernst relationship will be written in the form
(
=
(E)’’’
exp{g[Eoj - E ] ) =
where Eh;is the half-wave potential that each Oj/Rj couple would display in the absence of the other and of kinetic complications, R and Tare respectively the gas constant and the temperature, and g; is a potential-dependent parameter defined by eq 2.5. The behaviors of g, and g2 are sketched in Figure 1. Notice that gl and g2 have exactly the same shaped dependence on potential: they are simply translated along the potential axis relative to each other such that g, = at Ehj. We shall let k equal the sum k l + k2 of the two rate constants and also define a quantity g by (2.6)
Figure 1 includes a graph of g. Again g is simply a translation of curve gl or g2 along the potential axis, this
Electrochemlcal Reduction of an Isomeric Pair
time to a position such that g = given by
1/2
The Journal of Physical Chemistty, Vol. 87, No. 14, 1983 2495
a t the potential Eh
(2.7) If the 0 1 / R 1 couple is defined as the one with the more negative half-wave potential, as we shall find convenient, it follows that (2.8) g2 2 g 2 g1 a t all potentials, and for all magnitudes of k, and kz. An important role in the following development is played by a dimensionless parameter y defined by 2g1g2
1=
(2gz - 1)kz - (1 - 2gi)ki
(2.9) k2 + kl At sufficiently positive potentials y -1 as a limit, while at very negative potentials y 1. At the potential at which g = 2g1gz [which necessarily lies between Eh2and Ehl],y is zero. A graph showing a typical behavior of y as a function of potential is included in Figure 1. The individual Faradaic currents il and i2 are not, of course, experimentally accessible. Only their sum, given by eq 2.4 as
y=T--
--
i = Xij = - nAFD ( X
2)
Flgure 2. Current vs. potential curves for the cases kt = 0 and ki = 03 when D,i’2cbl= D 2 1 ’ 2 ~and b 2 k l / k p= 19.215. The current scale has been normalized by division by n A F ( D l ” 2 ~ b4-l D 2 1 ~ 2 c b 2 ) / ( r t ) ’ / 2 .
The assumed Nernstian behavior of the two electroreduction processes ensures that the equilibrium 0 1 + R2 R1+ 0 2 (2.17) is attained at the electrode surface, though not elsewhere in the solution, at all times and at all potentials. The equilibrium constant of (2.17) may be evaluated from eq 2.5 as
(2.10)
may be measured. Here and henceforth X indicates the summation of j = 1 and j = 2 terms. We seek a solution of the electrochemical problem expressed by eq 2.2,2.3, 2.4, 2.5, and 2.10, together with the initial conditions
(2.18)
O
(2.12)
and the asymptotic boundary conditions coj
Cbj
cRj-0
x
x - 0 3
03
all t
This system differs from a classical problemB only in there being two current components. Because the individual reduction currents are additive, the total current is given by
An example of the dependence of current on potential is shown as the leftmost curve in Figure 2. The currentpotential curve consists of two “wavesnwhich are more or less merged according to the magnitude of n[E, - E,]. Each wave has a height of nAFcbj(Dj/d)’fZand displays the typically Cottrellian t-1/2dependence upon time.
-
Moreover, because equilibrium 2.17 is always established at the electrode surface, it follows that equilibrium 2.19 is attained at the eleectrode surface only, that is cBo2= KOcBOl
k,
-m
k2
(2.23)
under these conditions, despite the fact that 0 1 and 0 2 do not isomerize directly. Hence when kl and k2 are both infinite, the electrochemical system corresponds to the reduction of one equilibrium mixture to another
c R’ (2.24) 02)
(26) See ref 2, p 158.
+
CR2
An exact treatment of this situation is presented in Ap-
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Bond and Oldham
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983
pendix A and shows that the current is given by
This expression corresponds to a current-potential graph that displays a single wave whose position along the potential axis depends on the kp/kl ratio, an example being shown in Figure 2. When k is neigher zero nor infinity, the electrochemical problem is again soluble, but with much greater difficulty. The mathematics involved has been relegated to Appendix B, in which it is proved that the current is expressed by
nAFg [V- exp(-kt) - V+] + -CDj112cbj(2.26) (7rt)lJZ where
v,
= =4( 27
5
Adequately to span the range of possible values of the kl/kz ratio, we have selected the values kl/kz = 0, exp(-4), 1, exp(4), exp(8), exp(l2), and m (3.4) for detailed study. As explained in the previous paragraph, the selection of a value from (3.4) implies fixing the value of KO and thereby making a statement about the relative thermodynamic stabilities of 01 and 02. That is, selecting kl/kz fiies Gof(Ol)- GOf(02) where G”f(0j) is the standard free energy of formation of 1 mol of O j as a solute in the electrolyte solution. The various implications of selecting specific kl/k2 ratios are detailed in Table I. The table also serves as an index to the diagrams contained in sections 4-6. The shape of the current-potential curve is determined not only by the kz/kl ratio but also by the kt product. For each k2/k1 ratio, we have selected representative values from the list kt = 0, 0.1, 1, 10, 100, 1000, lo4, lo5, lo6, and
[l f 712)
(2.27)
the 4 function being discussed in Appendix C. Gratifyingly, eq 2.26 reduces to eq 2.16 when k = 0 and to eq 2.25 when k = a. Unfortunately eq 2.26 is sufficiently complicated that its implications are opaque. However, if we abbreviate Vexp(-kt) - V+ to V, the equation may be recast as
(3.6)
and, using the computational methods discussed in Appendix C, constructed current-potential curves that are presented later as Figures 3-10. Of course the fact that O j may be thermodynamically unstable with respect to OJ presents no barrier to its being used as an electrochemical reactant. There are abundant examples of kinetically stable but thermodynamically unfavored isomers. Often it is unknown whether a particular isomer is the stable or labile form. 4. Reduction of 01
which shows that the current in the general case is a linear combination of the k = 0 current, given by eq 2.16, and the k = current, given by eq 2.25. This is not to say that the current-potential curves can be obtained by simply “mixing” the two graphs shown in Figure 2, because the “mixing parameter” V is itself a function of the potential. Moreover, V is also a function of time, indicating that the Cottrellian t-1/2dependence no longer holds. Note that our model has assumed the absence of any second-order homogeneous reaction, such as R1 + 0 2 01 + R2.
-
3. Wave Separation The theory we have developed in the previous section applies whatever the separation Eh2- Ehlof the half-wave potentials. The remainder of the article, however, will be restricted to cases in which this separation is large enough that, in the absence of kinetic complications, the two reduction waves would be quite distinct with a clear “plateau” evident between them. Approximately 6RT n[Ehz- Ehl]> -= 0.15 V (3.1)
F
satifies this criterion. In the examples used in this article we have taken 8RT = 0.20 v n[Ehz- Ehl]= 7 (3.2) In view of relationship 2.18, this choice is equivalent to K = 3.4 X Moreover, eq 2.21 then implies the relationship
’ k2
($)”‘KO
exp(8) = (3 X 1OS)Ko
(3.3)
between the rate constant ratio and the equilibrium constant of isomerization (2.19).
The theory developed in section 2 permitted a consideration of the reduction of any mixture of the isomers 01 and 0 2 but, in practice, one would usually have one isomer only present in the electrolyte solution or an equilibrium mixture of the two. In this section we deal with the case in which only the isomer 01 is present. This isomer, it will be recalled, is the one that would reduce at the more negative of the two half-wave potentials in the absence of kinetic complications. When kl is zero or much less than kz, the reduction of 01 is uncomplicated and yields a single Nernstian wave at Ehl. When kl and kz are comparable, the situation is similar but there is now some slight dependence of the position of the wave on the magnitude of the kt product. Figure 3 shows an example. When kl exceeds k2, the position of the reduction wave of 01 depends markedly on kt, as exemplified in Figure 4. The wave shifts toward positive potentials as kt increases because the isomerization reaction 01 + ne R1 2 R2 (4.1) becomes increasingly effective in removing the reduction product. Just detectable on careful inspection of Figure 4 are the steeper slopes of the kt = 10 and kt = 100 waves. The variation in steepness is clearer in Figure 5, which relates to a kl >> kz situation. When analyzed in the usual way, that is in terms of the gradient of In (id - i ) / i vs. E , the magnitudes of the slopes of the waves in Figure 5 are as listed in Table 11. Note that the “log plots” are not perfectly linear [except when kt = 0 or -1 and the tabulated values represent interpolated values at the half-wave potentials. Table I1 confirms that the slope of the current-potential curve is Nernstian in the two limits but is decidedly “super-Nernstian” at intermediate values of kt. This behavior is easily understood. The increase in current with
Electrochemical Reduction of an Isomeric Pair
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983 2497
Flgure 3. Current-potential curves for the case k , = k, and c b p= 0, when kt = 0, 1, or m. The current scale has been normalized by division by nAFcb,(D1/7rt)”2.
I
I
Flgure 4. Current-potential curves for the case k , = 56.60k2 and c b 2= 0, when kt = 0, 1, 10, 100, or m. Current normalization as in Figure 3. I
i
1
Flgure 6. Current-potential curves for the case k , = 162800k2and cbp= 0,when M = 0,I, IO, 100, 1000,io4, io5, lo6, atxi m. Current normalization as in Figure 3. No significant changes in shapes occur for larger values of the k , l k , ratio, or even for k , = 0.
c
Flgure 7. Current-potential curves for the case k , = 2981k, and c b l = 0 when kt = 0, lo5,and m. The current has been normalized by division by nAFc b2(D2/at)1’2.
and, in the potential range in which R2 is electrochemically inert, corresponds to the classical E,Ci case, about which much electrochemical literature exists.27
/ / /
5. Reduction of 0 2
I
1
Flgure 5. Current-potential curves for the case k = 2981k, and cbp = 0, when kt = 0, 1, 10, 100, 1000, lo4, lo*, and 00. Current normalization as in Figure 3.
increasingly negative potential in a normal Nernstian reduction results from a competition between the ne (Ol)s (RlP cathodic process and the reverse anodic process. The former is not directly affected by the isomerization of R1 to R2 but the latter is hindered, and hence the current increases with negative potential more than in a normal Nernstian process. When k2 is zero, or much less than kl,the situation is as exemplified in Figure 6. The reduction wave of 01 lies somewhere between Ehland EhZ,and is generally superNernstian. The reaction scheme in this case is effectively
-
+
ne
01 == R1- R2
(4.2)
In this section we treat a solution that inititally contains only 0 2 , the isomer that reduces at the more positive of the two half-wave potentials, When kz is zero, the reduction of 0 2 occurs in a simple Nernstian wave, centered at Ehz.The situation is not much changed when k z > Itl but with only minor changes also applies to the kl = 0 case. All kt values generate current-potential curves with either wave or peak shapes. Notice that in the vicinity of (Ehl + Eh2)/2,all curves are approximately horizontal. By inverting eq B.12 of Appendix B under the conditions p
-
Electrochemical Reduction of an Isomeric Pair
The Journal of Physical Chemistry, Vol. 87, No. 14, 1983 2400
t-0
I
5
/ //f
I
I
Figure 10. Current-potential curve for the case k , = 0.01832k2and cbl = 0 when kt = 0,0.1, 1, 10, 100, and m. Current normalization as In Figure 7. No significant change in curve shapes occurs for smaller values of the k , / k 2 ratio, or even when k l = 0.
= 0 = g,, y = 1 = g,, one can show that the approximately
F m e 11. Cwent-potentlal curves for the case of an initial equilibrium mixture of 01 and 0 2 with k l = 2981k,. Wlth the additional assumption D , = D2, this corresponds to cb1= cbp. Current values for kt = 0, 1, 10, 100, 1000, 10000, and m are graphed. The current has been normalized by dhrlsbn by nAF(D,”2cb, D,1’2cb2)/(?rf)1’2.
+
potential-independent current is given by
- -
in this region. For kt > 10, these currents are miniscule but nevertheless the conversion 0 2 R2 R 1 - 0 1 is proceeding in what might be termed an electrodecatalyzed isomerization; we discuss this phenomenon in a later article. 6. Reduction of an Equilibrium Mixture We have stipulated that 0 1 and 0 2 do not equilibrate or, more precisely, we have assumed that any interconversion of 0 1 and 0 2 is negligibly slow on the time scale of the electrochemical experiment. It could nevertheless be the case that 0 1 and 0 2 do slowly equilibrate in the electrolyte, or that the method of synthesis of the compound yielded an isomeric mixture. For either of these reasons one might commonly encounter situations in which the bulk concentrations of the solutes obey the relationship
Such a mixture will contain insignificant concentrations of 0 2 (or 0 1 ) if k l / k 2is much less than (or much greater than) 3000 and the results of section 4 (or section 5 ) can be taken over unchanged. Reference to Table I shows that, of the seven k l / k 2cases there considered, it is only the fifth that must be specially treated here. The result of this treatment is shown in Figure 11. It corresponds to exactly equal bulk concentrations of 0 1 and 0 2 . Because of the interconversion of R1 and R2, it is only in the kt = 0 case that Figure 11 represents the average of Figures 5 and 7. Notice in particular that the negative shift of the curve, with increasing kt, in Figure 7 does not occur at all in Figure 11, even at the most positive potentials. The negative shift was caused by oxidation of R1 but, with excess 01 now present, this oxidation does not occur to any extent. 7. Time Dependence
It was noted in section 2 that, in general, the Cottrellian behavior characteristic of reversible potentiostatic currents is destroyed by the interconversion of the reduction products. The same data that were used to generate Figures 3-10 may also be used to demonstrate the time
001
01
I
100
IO
103
lo4
\
Figure 12. Logarithmic current-time curves for the cases of 0 1 or 0 2 initially present alone, with k , = 58.60k2 (as in Figures 4 and 8). The constant potential is (Eh, Eh1)/2. The curve &beled C represents Cottrelllan behavior. Currents have been normalized by division by nAFcb,(D,4/?r)1‘2.
+
dependence. The curves shown in Figure 12 were generated in this way, augmented by new data. Notice the Cottrellian behavior (straight lines in Figure 12 with slopes of -l/,) in the short- and long-time limits, with deviations at intermediate times when t is comparable with l / k . In the case of 0 2 , it is doubtful if the mild deviation from dependence toward a steeper dependence could be detected experimentally. This is not the case when the starting species is 0 1 , however: the deviation is marked. In fact, in the region 1 5 k t 5 10 the current is virtually time independent. An approximation to this constant current can be obtained by inverting eq B.12 of Appendix B under the prevailing conditions [p = 2q = ~ ~ A F ~ , C ~yD=~-1; ’ / kt ~ ;L 11. One obtains
8. Conclusions Our calculations show conclusively that reductions experimentally indistinguishable from simple reversible waves may be obtained even when the reactant and product are structurally dissimilar. For example, some of the curves in Figures 4-6 relate to this condition as does any system fitting scheme (1.9).Some other waves in these figures have super-Nernstian slopes: this is in contradiction to the philosophy that structural change correlates
2500
Bond and Oldham
The Journal of Physical Chemistty, Vol. 87, No. 14, 1983
with less-than-reversible behavior. However, the electrochemical consequences of the "square" mechanism may be more profound than a mere distortion of the wave slope, as Figure 10 with its peaks and plateaus illustrates. Conversely, we have uncovered circumstances in which reversible electron transfer with no net structural change can lead to sub-Nernstian behavior. Figure 7 shows an example. Again, however, more complex behavior may result from the square scheme, as Figure 8 illustrates. Our analysis has related to potentiostatic electrochemical techniques, but we believe that our findings can be extrapolated to other techniques such as the popular cyclic voltammetry, since our theory obviously bears a close relationship to the so-called ECE and ECE reaction schemes already examined in an entirely different context by Sav6ant,28Feldberg,%and others with respect to to cyclic voltammetry and related techniques. Unfortunately, far more complex mathematical solutions are required to solve the appropriate equations under conditions pertaining to cyclic voltammetry. Our overall conclusion is that, without a knowledge of the mechanism, no inference concerning structural relations between reactant and product can be drawn from current-potential curves. Moreover, the drawing of mechanistic conclusions from voltammetric data alone is evidently a very dangerous procedure. Acknowledgment. The granting of leave by Trent University and the hospitality of Deakin University are recognized with gratitude. This work was supported by the Natural Sciences and Engineering Research Council of Canada, Deakin University Research Committee, and the Australian Research Grants Committee. Appendix A The overall transportlreaction scheme that we consider here may be diagrammed as follows: (OUb
(02Ib
(01)'
Dz
KOJ
S
A
(R1f
iPR K ~ 1J
1
(02)'
(RI~
ne
e
(R2f
D
(-4.1)
(R2f
where all reaction steps are infinitely fast and all transport steps obey Fick's laws. With bars denoting the result of Laplace transformations with respect to t, the transform of eq 2.2 is d2eoj Dj= scOj - cbj dx where s is the dummy variable and the initial condition 2.11 has been inserted. Subject to the requirement of condition 2.13, the solution of (A.2) is
{
toj= Cbj S - (constant) exp -x
(;)"I
(A.3)
and therefore
( 2 )($[ =
j = 1 or 2 -
(A.4)
If we turn now to the reduced species, it is evident that the ratio of the concentrations of R2 and R1 will equal the
constant KR at all times and at all points in solution, and not only at the electrode surface as in (2.22). Hence defining we can replace eq 2.3, 2.12, and 2.14 by
cR-0
allx>O
(A.7)
(A.8) These last three equations may be Laplace transformed and processed by operations similar to those used in the previous paragraph, to arrive at allt
