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Kinetic parameters from steady-state voltammograms at microdisc electrodes
79
J. Electroanal. Chem., 270 (1989) 79-101 Elsevier Sequoia S.A.. Lausanne - Printed
in The Netherlands
Kinetic parameters from steady-state microdisc electrodes
voltammograms
at
Keith B. Oldham and Janice C. Myland Department
of Chemrstr).
Trent Unroersrt.v, Peterborough,
Ontarro. K9J 7B8 (Cunudu)
Cynthia G. Zoski and Alan M. Bond Depurtment (Received
of Chemrcai and Ana!vtrcul Scrences. Dealrm Unwer.rrty. Geelong, Vwtorru 3217 (Austraba) 21 March
1989)
ABSTRACT Exact methods of determinmg the kinetic parameters a and k”, as well as the thermodynamic parameter E O, from steady-state voltammograms at mIcrodIsc electrodes are presented. These methods rely on employing a range of mlcrodw radii in order to scan the reversible, quasi-reversible or irreversible kmetic regimes. The experlmentally determined quartlle potentials E, ,4, E,, ?. and E, ,4 are either graphed or used to locate pomts on the so-called “kmetic Indicator diagrams” from which a. k O. and E” may be determined. Our procedures are not based on the usual assumptions of “uniform accessibility” or equal diffusion coeffu.zients.
INTRODUCTION
Two parameters, the transfer coefficient (Y and the standard heterogeneous rate constant X-O, are used to characterize the kinetics of simple electron transfer reactions such as the reduction O(soln)
+ n eP -j R(soln)
(1)
the In addition, the standard potential E o must also be known to characterize reaction fully. Voltammetric experiments may be classified according to whether the electron or “irreversibly”. This classifitransfer step occurs “reversibly”, “quasi-reversibly”, cation of a particular experiment determines which kinetic parameters, if any. are accessible, as detailed in Table 1. The basis of the classification, for conventional transient voltammetric experiments, involves a comparison of the ratio Do/(kO)’ with some characteristic time 7 [l]. Here Do is the diffusion coefficient of the electroreactant and, depending on the nature of the experiment, r may be a drop 0022-0728/89/$03.50
‘is 1989 Elsevier Sequoia
S.A.
80
TABLE
1
Classification
of voltammetrlc
regimes
and the kinetic parameters
which are experimentally
accessible
Regime
n
ho
reversible quasi-reversible irreversible
maccessible accessible accewble
maccesslble accessible accessible accessible maccesslble. though a parameter involving both h-O and E 0 is calculable
time, a pulse width, a reciprocal sweep rate u. One has Do/( k o )” cc
reversible
Do/(/?“)‘=
quasi-reversible
7:
Do/( k o )’ > 7:
frequency,
E”
or the quantity
RT/nFu
involving
a
(2)
irreversible
As is evident from Table 1, experiments in the quasi-reversible regime are the most advantageous for kinetic studies. Because diffusion coefficients have magnitudes close to lo-9 m’ s-l and many interesting electrode reactions have standard rate constants of lo-” m s-l or greater, we see that characteristic times of one millisecond or less are required to attain quasi-reversibility. There are, however, a number of experimental difficulties with transient voltammetry on such timescales, notably interference from the large charging currents that flow under such conditions [2-41. A way around these difficulties is to espouse steady-state experiments. In the steady state there is no charging current whatsoever, and no characteristic time either. Instead, the role of determining the voltammetric regime is taken over by a characteristic length. X. dependent upon the dimensions of the electrode. It is the comparative magnitude of this length and the quotient Do/k” that is crucial [5]. One has Do/k”
<
reversible
Do/k”
= A:
quasi-reversible
Do/k”
>> A:
irreversible
(3)
For Do = 10e9 rn’ s-’ and k” = 10-j m SC’, one needs a characteristic electrode dimension h of order one micrometre to achieve quasi-reversibility. Recently there has been much activity and success in the fabrication of electrodes of micrometre and submicrometre dimensions [6-91 opening the door to the use of steady-state voltammetry [lO,ll] as a means of measuring electrochemical kinetic parameters. Perhaps the most easily constructed microelectrodes are those with the geometry of an inlaid disc [6-91, in which a flat circular electronically conducting disc is embedded in an insulator whose surface forms a geometric continuation of the electrode/solution interface. It is with such microelectrodes that this article is solely
81
concerned. For inlaid disc microelectrodes we may associate the characteristic length, introduced in the last paragraph, with the term 7ru/4 where a is the radius of the microdisc. Further, we define a dimensionless parameter K* by K0
=
hk o,‘D, = nk “a/(4Do)
(4)
so that in the light of eqn. (3) one has It”>l:
reversible quasi-reversible
K o 21: K”
(<
(5)
irreversible
1:
In theory, full reversibility and total irreversibility are conditions that are approached as limits, so that all experiments may be regarded as quasi-reversible. In practice, however, the useful range of quasi-reversibility embraces only about two orders of magnitude of the KO parameter, as shown by the classification K”
>
0.2 5 K”
<
40: K* 0.1:
effectively <
20:
usefully
reversible
quasi-reversible
effectively
(6)
irreversible
The purpose of this article is to propose practical and convenient methods by which the parameters LY,k *, and E o may be obtained from steady-state voltammograms recorded using inlaid disc microelectrodes of a variety of radii. Our methods are able to handle voltammetric data obtained from experiments conducted under quasi-reversible conditions, sometimes augmented by purely reversible or totally irreversible data. Only a limited range of k“ values is accessible. Table 2, which is based on the typical diffusion coefficient value Do = 1 X 10e9 m2 s-‘, lists the quasi-resmallest and largest disc radii that cause KO to fall within the “usefully versible” range specified in eqn. (6). The present state of the art of microdisc fabrication limits the use of discs smaller than about 0.25 pm in radius [9]. Large discs are easily made but the larger the disc, the longer it takes to achieve a steady state, and currents at potentiostatted discs of radii much larger than 25 pm take so long to reach constancy that convection often intervenes [ll]. Accordingly, the present methods will find greatest applicability in studying electrode reactions having standard rate constants in the range 10 m4 to lo-’ m SC’.
TABLE
2
The smallest k”/m lo-’ 10-z 10-j 10-4 10-5
s-’
and largest
disc radii that cause k * to fall wthin
the “usefully
a/pm 0.00250.25 0.015 2.5 0.25 25 2.5 - 250 25 -2500
quasi-reversible”
range
x2
In this work it is assumed that diffusion is the sole transport mode, conditions having been selected to obviate convection and migration. Of course, forced convection can be avoided by not agitating the solution but, as mentioned earlier, natural convection may obtrude unless the disc electrode is small enough that a steady state is reached sufficiently rapidly. To prevent migration, supporting electrolyte will have been added to the solution phase. A serendipitous feature of voltall~metry at microelectrodes is that for certain electrode reactions (those that increase the ionic strength at the interface) far less supporting electrolyte is needed than is the case for studies with macroelectrodes [I?-181. VOLTAMMETRIC
SHAPES
Steady-state voltammograms may be recorded in a point-by-point fashion by performing a series of chronoamperometric experiments, each at a slightly different potential [19]. More conveniently, they result from applying a potential ramp to the microdisc electrode, provided that the sweep rate is slow enough that the voltammogram retraces itself when the direction of the ramp is reversed [3,8.20.21] (though a means of circumventing this difficulty has recently been suggested 1221). A third method would be to record the ultimate potentials reached during a series of chronopotelltiometric experiments each at a slightly different current (not exceeding the limiting current), while a fourth could employ a slowly increasing current ramp. It is a valuable characteristic of steady-state voltammetry that the resulting current-voltage curves are totally independent of the details of the method used to obtain them [tl]. Figure 1 shows the typical shapes of reversible, quasi-reversible and irreversible steady-state voltammograms that result from experiments at microdisc electrodes. Note that the same limiting current 15,231 d, = 4~~Do~~U
(7)
is attained irrespective of the degree of reversibility, but that the slope of the voltammetric wave diminishes as one proceeds from reversibility through quasi-reversibility to irreversibility. The three curves in Fig. 1 correspond to reductions of Do) three substances (all of equal bulk concentrations c:, and diffusion coefficients that share the same values of a! and E* but differ in their standard rate constants. It is evident from the figure that the waves not only become less steep but also shift towards more negative potentials as k” decreases. Ret~ersible waves The shape of a reversible
steady-state
voltammogram
obeys the simple
equation
PI i=i,/fl+(Do/L),)exp[njc(E:-E’)/RT]]
(8)
83
0.50-
0.25-
I
I
I
200
I
-200
0
I
-4c
Fig. 1. The shapes of reversible (k o = co), quasi-reversible (k o = 1.6 Do/au) and effectively irreversible (k” = 0.4 Do/m) steady-state voltammograms. The parameter values (Y= 0.5, n =l.O, Do/D, =l.O and T = 298.2 K were assumed.
relating current i to potential E. From this it follows that the difference the standard and half-wave potentials satisfies the relationship n(E”
-E1,2)
= CRT/F)
ln(D,/D.)
between
(9)
Because the diffusion coefficient D, of the product of the electrode reaction rarely differs significantly from that of the reactant, the half-wave and standard potentials are virtually identical for reversible electrode reactions. A useful way of quantifying the slope of a voltammetric wave was pioneered by Tomes [24] and involves measuring the difference between the one-quarter-wave potential and the three-quarters-wave potential. From eqn. (8) one can show that, for a reversible n-electron reduction wave, this potential difference is given by n ( E,,4 - E3,4) = (RT/F)
ln(9) = 56.5 mV
(10)
and is not influenced by any disparity between Do and D,. Notice that, as presaged in Table 1, no information about (Yor k” is contained in eqn. (9) or (10). The millivoltage given in eqn. (lo), and in every potential difference cited subsequently in this article, relates to 25.0 o C. For other temperatures T, the stated millivoltages (including those marked on the axes of diagrams) should be multiplied by T/(298.2 K). Quasi-reversible
waves
The shape of quasi-reversible steady-state voltammograms from microdisc electrodes has been the main subject of several publications from these laboratories
84
[5,11,25]. In terms of a parameter
8, defined
by
8=1+(D,/D,)exp[nF(E-E”)/RT] and a second parameter
(11)
K,
defined
by
K=K”exp[-OLnF(E-Eo)/RT] the current-potential
(12)
relationship
for a quasi-reversible
in recent work by Michael as
This equation has been verified independently Equation (13) can be written more succinctly 1 --I= 40
wave is given by [25]
et al. [26].
2(h + 1) 3h(2h
(14)
+a)
where k is a new parameter, ,=%=%[I+2
proportional
to the
exp{$(E-E”)i]
while q is simply
K$
product,
and defined
by (75)
exp(s(E-E”))
the height of the voltammetric
current
q = i/i,
relative
to the plateau (16)
Analogues of eqns. (9) and (10) do not exist for quasi-reversible conditions and one must be content with numerical or graphical equivalents. In experimental practice, one rarely has accurate data on the diffusion coefficients of both 0 and R, so it is customary to assume Do = D,. We shall follow this custom by henceforth ignoring the (D&D,) multipliers in eqns. (11) and (15). With this simplification, these two equations may be combined into 21c08=3rk(d-l)a
(17)
In Appendix A, we discuss the minor modifications that are necessary if the equality of Do and D, is not assumed. Our interest focuses on the three quartile potentials: E,,4, E,,z and EX,4, which we represent collectively by E,, with q = l/4, l/2 or 3/4. For given values of q, a the voltage n( E, - E “) by first using an iterative and KO, one can determine method to solve the equation 2(k+l) 3k (2k + T)
Ii
for k and then substituting $(E,-EO)=Lni[q+
a 1+
2(k+l) 3k(2k
+ T) I
'-a_
2K”
3n.q
(18)
this value into ~(2hh=1n))]-~-1)
(19)
85
Equation (18) is derived by eliminating 6 between eqns. (14) and (17). while eqn. (19) follows directly from eqn. (14) and the definition of 8. We employed the procedure of the last paragraph to calculate values of n( E O E,,,), n(E” - E,,4) and n( E o - E,,z) at round values of a: and log. Subtraction of n(E” -I?,,,) from n(E’ - E,,,) then generates the potential difference - E3,4) separating the one-quarter-wave and three-quarters-wave potentials. n(E,,, Because values of n(E * - E,,?) are also known at the same round values of (Yand log{K” ), Figs. 2 and 3 could be constructed. These diagrams are small-scale and large-scale versions of plots of n( E o - E,,,) versus n(E,,, - E,,,,) in which “contour” lines link points of equal (Yand points of equal logfrr 0 ). They show how the slopes [as represented by n( E,,4 - E3,4)) and positions [as represented by n(E” - E,,2)] of quasi-reversible steady-state waves at microdisc electrodes depend on cf and log( Ko )_ To locate a point on either Fig. 2 or 3 requires that one know a value for the standard potential E o in order to calculate the ordinate n( E o - E,,z). Unforvoltammograms tunately, the determination of E” from a series of steady-state recorded at microdiscs of various radii may not always be possible. Thus, it may be convenient (and experimentally is always possible) to measure instead the change in half-wave potential E,,,2 with disc radius a and thereby calculate one or more values of the n
dE,,,/d
log a
(20)
quantity. These experimental values may then be used in conjunction with Figs. 4 and/or 5, which show how the~quantity (20) is related theoretically to the Tomes cx and log(~ o ). Appendix B separation n (E,,, - E3,4) and the kinetic parameters describes the construction of Figs. 4 and 5. These diagrams are small-scale and large-scale versions of plots of nd E,,,/d log a versus nf E,,, - E3,_,) in which contour lines again link points of equal (Yand log( K o ) as in Figs. 2 and 3. Figures 4 and 5 show how the effect of the disc radius on the half-wave potential [as represented by dE,,,/d log a) and the wave slope [as represented by n( E,,,, - xT;,~)] of quasi-reversible steady-state voltammograms depend on (Y and log(K”). We call Figs. 2-5 “kinetic indicator diagrams”. Irreversible
waves
Total irreversibility corresponds to the reduction wave being displaced negatively from the standard potential to such an extent that, at all points on the escarpment of the voltammogram, the term exp[nF( E - E * )/RT] is negligible in comparison with unity. It follows from eqn. (11) that 8 is effectively equal to one. Hence eqn. (13) reduces to the simple result
(21) for an irreversible
wave.
86
v.
7
,\$ /-
1’ . ./’. . / .
,
. 1-Y.
.o\
/:
.
.
*
.
/
.
..’ /
+
/
’ /: c
;/
_.+y
. . . __./
.,I-
. . .
.
.A..
/
-.
/_,‘.
.
.
/Q~K
fi
. #
c
c_
. a= 60
80
i
.oo
5z&,mV I
20
_w+
,,,A- 0 2. . . .. 4: Z0~T.z-L _ A__-____;_
I i
1_
114
140
160
f
5/4
I
180
2013
Fig. 2. A small-scale kinetic indicator diagram which shows the relationshlp between n( E o - E,,z) n (E, ,4 - E,,,) at constant a or log( K o ) for steady-state voltammograms at microdlsc electrodes.
Setting i/i,
equal to one half in eqn. (21), one finds
n(E" - E,,z)= (RT/c&)[0.06973
- ln(ic’=)]
K =
and
1.072 so that (22)
when eqn. (12) is invoked. Similar expressions involving E3,4and E,,4emerge from setting i/i, equal to 3/4 or l/4 respectively in eqn. (21). Subtraction of these two latter expressions leads to [25] n(
E,,4- E3,4)= 2.342RT/aF= (60.2/a)mV
(23)
87
4c
35
3c
25
20
15
10
,0-g . . . . .
S
9_
I
60
Fig. 3. The expanded
65
70
75
inset of Fig. 2.
Hence, as reported in Table 1, a value of (Y is easily accessible from an irreversible steady-state voltammetric wave at an inlaid disc microelectrode simply by measuring the separation of the one-quarter-wave and three-quarters-wave potentials. For an irreversible wave, this separation is dependent on neither the disc radius a nor the rate constant k”. Thus, experimental evidence for an irreversible wave is a constant Tomes potential difference as the disc is diminished in size. Returning to eqn. (22), one finds 8.0 mV + 25.7 mV ‘F ln(k”) ln(Do/a)=nE”+(24) n-G/, + ~ (Y Ly
88 >:
16tI- .<
:
:
:
:
1.
“:....:““:....:..
n
.
.‘(5
:
14tI- ._
I&$(?$?7 :F : /
. . . . . . . . . . . . . . . . . .
F
;
:
.
F
12cI\
*
*
Y) h.
L 0-
1OC I-
.,r,!.
./r..
./.
. .
.,.-r-i.
8C
6C 9
4c
I<
_--. . . . . . .
Ly-T-7-y . . . . . . . . . . .._..............
20 IFig 0
J
‘Ei0_
Fig.
P
4.
d&,/d microdisc
A
1
1
I
I
I
80
100
120
5
small-scale
kinetic
indicator
diagram
1 ’ . n CE1/4-E3;41 /III\ J I
I
140 whxh
160 portrays
the
I
180 relationship
1
200 between
n
E,,,) at constant a or log( I(’ ) for steady-state voltammograms at log(a) and n(E,,,electrodes. The unlabelled log(K “) lines correspond to -0.4, - 0.6, - 0.8, - 1.0 and - co.
which explains the allusion in Table 1 to the accessibility from an irreversible experiment of a parameter involving both k ’ and E ‘, though neither of these quantities is separately accessible. METHODS
OF DETERMINING
KINETIC
PARAMETERS
The principle of using steady-state voltammograms at microdisc electrodes to measure the kinetic parameters, LYand k O, as well as the thermodynamic parameter
4c
35
30
2s
20
15
10
5
0 c Fig. 5. The expanded
inset of Fig. 4.
E O, of reaction (1) relies on employing a range of disc radii u. In this way, one is able to scan a range of values of the dimensionless parameter ICO (because this equals vrk”a/4Do). We have already remarked that the usable range of disc radii is rather limited and a similar limitation consequently afflicts K”. As portrayed in classification (6), the total spectrum of KO values embraces reversible, quasi-reversible and irreversible voltammetric behaviours. The small accessible range of KO may intersect this total spectrum in a variety of ways, as illustrated in Fig. 6. If the accessible range of KO values lies entirely within the reversible region, as illustrated by Case 0 in Fig. 6, or entirely within the irreversible region, as illustrated
90
Case VI ,*YF*YT*Y/-. y///‘.“““‘y/A / Irreversible \
I
-3
I
-2
Case II
Case IV
Cose 0
v”“““““/g v/,““““y///// /Quasireversible Reversible ,\ / 4 Case V Case III Case I
I
-1
I
II
I 1
I
2
log (K”)
I
>
3
Fig. 6. The “spectrum” of voltammetric expenments, showmg how the Irreversible, quasi-reversible and reversible kmetlc regimes reflect log( Ko ). The seven cases dlagrammed are discussed m turn in the text.
by Case VI, then only one parameter is determinate. However, for each of the other five cases diagrammed in Fig. 6, the experimentally determined quartile potentials (Y, k o (Q/47 El/z. and E,,,) may be employed to determine the three parameters must be employed in each case. All these and E”, though a distinct procedure procedures, which will now be discussed in turn, make use of one or several of the kinetic indicator diagrams, Figs. 2-5. Other approaches to this problem have been considered recently by Galus et al. [27] and by Aoki et al. [28].
Method I: reversible/near-reversible
Case I in Fig. 6 shows a situation in which the accessible range of KO includes portions of both the reversible and quasi-reversible regimes. One can identify this situation from the fact that the largest disc will show a n( E,,, - E,,,,) separation of 56.5 mV within experimental error, as predicted by eqn. (lo), whereas smaller discs will generate voltammograms whose one-quarter and three-quarters potentials are separated further. In this case, the half-wave potential developed by the largest disc will obey eqn. (9) and will therefore equal the standard potential E o (recall that we are assuming that the two diffusion coefficients are equal: see Appendix A if this assumption is rz( E, ,,4 - E3,4) not made). Now knowing E O, one may calculate the coordinates and n(E” - E,,2) of a point on kinetic indicator diagram Fig. 3 for the voltammogram from each of the smaller discs. The location of such an experimental point on the kinetic indicator diagram permits the simultaneous estimation of (Y and of log( K o ). Thence, via the relationship k” = 4D,K”/Ta
(25)
the standard rate constant k” may be determined. Each experimental voltammogram from a smaller disc yields a distinct point on the kinetic indicator diagram. The points should lie on, or parallel to, lines of constant cll, but will correspond to various values of log( K”) because this parameter incorporates the variable disc radius. Nevertheless, use of eqn. (25) should lead to a unique k o value.
91
Method II: near-reversible/quasi-reversible As illustrated in Fig. 6, Case II corresponds to a situation in which the accessible values of KO do not enter the reversible region, but lie close to it. That one has encountered this situation will be evident from the fact that the experimental Tomes differences, n ( E1,4 - E3,4), will all exceed 56.5 mV but will approach this value for the largest discs. The procedure that we advocate in this case involves an extrapolation into the reversible region, yielding a value of the standard potential E O. In Appendix C (and elsewhere [lo]) we demonstrate that, in this near-reversible regime. a graph of nE,,, versus l/a will yield a straight line of slope - 2RTD,,/Fk” and intercept nE O, thus providing a very convenient method for determining both k o and E O. Alternatively, in the region of near-reversibility a linear relationship exists between n ( El,z -E”) and n(E,,, - E3,4) for discs of differing radii, as predicted by eqn. (ClO) and illustrated in Fig. 7. The extrapolation indicated in Fig. 7 is used to find EO, while the slope l/(3”-’ - 3’-,) of the line can be used to provide a value of (Y. The values for the eight data pairs were calculated as described previously and are reported in Table 3 along with the corresponding disc radii. The line in Fig. 7 has been drawn in accordance with eqn. (ClO). The points corresponding to near-reversibility lie on, or close to, the line whereas the less reversible points show a mild departure from linearity. Nevertheless, a regression line drawn through all eight points would give a value of (Y(calculated from the slope) which is in error by only 3%. An alternative method of determining (Y is from the kinetic indicator diagram Fig. 3. Knowing E o and k” (and hence KO ) from a graph of nE,,, vs. l/a, one
nE,,JmV
-llO-
I '
56
I 58
I 60
I 62
nK,,4-E,,,) /mV I 64
I 66
Fig. 7. Graph of half-wave potential versus the difference between three-quarters-wave potential for the hypothetical electroreductlon
the one-quarter-wave potential detailed m Table 3.
and the
92 TABLE
3
Calculated halfwave potenttals and Tomes potential differences E”=-lOO.OmV, k”=O.Olms-‘, D,,=D,=1.0X10-9m’ microdisc electrodes of the stated radit. Values of nE,,, and difference was calculated are also tabulated
for a hypothettcal electroreductton with sm’,T=298.2K, n=land oc=0.5,at nE,,, from whrch the Tomes potenttal
a/pm
nE,,jZ/mV
nEt,,/mV
nE:,.,/mV
n(E, ,4 - .%,)/mV
0.51 0.64 0.80 1.01 1.21 2.02 3.20 8.03
-
-
-
66.4 64.5 63.0 61.7 60.7 59.2 58.2 57.2
108.3 106.8 105.5 104.5 103.6 102.4 101.5 100.6
16.6 15.1 75.0 14.4 73.9 13.2 72.7 72.1
143.0 140.2 138.0 136.1 134.6 132.4 130.9 129.3
can easily locate a series of points along one of the (Ycontour lines, thus providing a measure of (Y.The value of (Yso obtained may be compared to that calculated from the slope of a plot resembling Fig. 7. Method III: quasi-reversible In Case III, experimentally accessible values of ICO lie solely in the quasi-reversible regime. Experimental confirmation that one has encountered this situation will be reflected in the n( E,,, - E,,,) potential separations, which will not approach a limit at either the smallest or the largest discs studied. In this case, one must rely solely on the kinetic indicator diagrams. Since E o will generally not be known, one resorts first to Figs. 4 and 5 which make use of the shift of n&,2 with electrode size in relation to n( E,,4 - E,,,,). This permits values of (Y simultaneously. Values and log( K o ) [and hence k o via eqn. (25)] to be determined of OLand log(~O ) in addition to n( E,,, - E,,,,) provide the necessary information for locating a point for each voltammogram on the Fig. 2 or 3 kinetic indicator diagram which relates n( E o - E,,, ) and n( E,,4 - E,,,, ). From a particular value a unique E o value may be found. of n(E” - E,,z) for each voltammogram, Method IV: near-irreversible As portrayed in Fig. 6, Case IV corresponds to a situation in which the accessible values of KO do not enter the irreversible regime, but lie close to it. One can identify which will increase this situation experimentally by the n( E,,4 - E,,.,) separations with decreasing disc size and approach a limit [the 60.2 mV/a irreversible limit] with the smallest discs. The procedure that we advocate in this case involves an extrapolation into the of each voltammogram irreversible regime. Measure the E,,4 and E,,, potentials and plot the quantity L = 0.3654 exp
n(El14- E,) 25.1 mV
- 3.472 exp
n(Q4 - -%I\ 25.7 mV
1
93
642-
Fig. 8. Graph of the experimentally calculated parameter L versus the measured one-quarter-wave three-quarters-wave potential separation for the hypothetical electroreduction reported m Table 3.
and
versus n( E,,4 - E,,,). Here E, is any constant reference potential in the vicinity of the voltammetric waves; it could, for instance, be the half-wave potential of one of the voltammograms. In Appendix D, we demonstrate that. in this near-irreversible linear and may be extrapolated to regime, such a plot will be approximately intersect the n( E,,4 - E,,,) axis at (60.2 mV)/u. This intercept permits a determination of a. while the standard potential is then calculable from the slope by using the result (25.7 mV) ln{ -(25.7 mV)(slope)/a} tlE” =nE,+ (27) To illustrate this procedure, we have prepared Fig. 8. Table 4 tabulates the values for the eight data points at the corresponding disc radii. The solid line of Fig. 8 was
TABLE
4
Data for the Fig. 8 graph of the experimentally determined parameter L (defined in eqn. 26) versus the one-quarter-wave and three-quarters-wave potential separation. Values of k o = 1 x 10e4 m s- ‘. D = 1.0 x 10m9 rn’ SC’. n = 1. a = 0.5. T = 298.2 K, E o = - 100.0 mV and E, = - 200.0 mV were assumed. The numerical value of the disc radii listed in the first column led to the remainmg four columns of data
a/v
L
fi (6
0.80 1.3 1.6 2.0 2.5 3.2 4.0 5.1
0.5422 1.310 2.003 3.009 4.411 6.260 8.537 11.12
119.8 119.0 118.2 117.1 115.5 113.2 110.3 106.6
,4
-
4,
)/mV
ME,,., -- E,)/mV 12.6 35.3 46.3 56.9 66.9 16.2 X4.6 92.0
n(E3, 4 - 107.2 - 83.6 -11.9 - 60.2 -486 - 37.0 - 25.7 - 14.6
-
E,)/mV
94
drawn according to eqn. (D12). The points corresponding to near-irreversibility lie on, or close to, the line while the more reversible points depart from the line of theoretical slope. A regression line drawn through all eight points leads to a standard potential which is in error by 4 mV and a value for the transfer coefficient which is 0.3% low. Now, knowing E o and (Y,one may calculate the coordinates n( E o - E,,,,) and - E3,4) of a point on kinetic indicator diagram Fig. 2 for the voltammogram n(E,,, from each of the discs. The location of each experimental point on the kinetic indicator diagram permits the determination of a unique value of log( K o ) and hence k o from relationship (25). METHOD
V: NEAR-IRREVERSIBLE/IRREVERSIBLE
In this last case, the accessible range of K a values includes portions of both the quasireversible and irreversible kinetic regimes. One can identify this situation experimentally from the fact that the smallest discs will have identical n ( E, ,4 - E3,4) separations, whereas larger discs will generate voltammograms whose one-quarter and three-quarters potentials are less separated. For the smallest discs, the n( E,,4 - E?,,) separation will obey eqn. (23), permitting cx to be calculated as (60.2 mV)/(nE,,, - nE,,,). One can then locate the appropriate (Ycontour on Fig. 2 and insert points corresponding to the larger discs at the intersections of this contour with the measured n( E,,, - E,,,,) abscissas. From each such point, values of n (E o - E,,,,) and log( KO) can be read off. From the first. E o is calculable using the measured E,,,, while from the second, k o may be found via eqn. (25). SUMMARY
OF METHODS
Here we summarize the methods advocated for the determination of the parameters E”, k o and cx from measured steady-state values of E,,4, E,,?, and E,,, obtained from microdisc voltammograms using a series of disc radii u. In this summary we make the simplifying assumptions that n = 1. T = 298.2 K and that the electroreactant has a known diffusion coefficient D equal to that of its reduction product. If no variation with disc radius, beyond what can be attributed to experimental scatter, is observed in El,?. and E,,4 - E,,, - 56.5 mV, then (Y is indeterminate, E o can be equated to an average E,,,, and k o exceeds 51 D/a m,n where a m,n is the radius of the smallest electrode. If E,,, - E3,4 equals 56.5 mV for the largest discs but exceeds this value for the smaller electrodes, E o may be equated to an average E,,2 for the largest electrodes. For each of the smaller electrodes, locate the point on Fig. 3 that corresponds to the coordinates E, ,4 - E,,, and E o - E,,?. All these points should lie along a line of constant CX,whence this parameter may be found. Estimate log( K o ) for each point and calculate ~DK~/~u; a constant value, equal to k O, should be obtained.
95
If all values of E,,4 - E3,4 exceed 56.5 mV but this value is approached as the disc size increases, then two methods are available. Both require that E o be found by extrapolation. In the first method, plot E,,,, versus l/a: the intercept at I/a = 0 is E” and k” is equal to (-51.4 mV)D/(slope). In the second method, plot E,,z versus E,,4 - E, 4; the intercept at E,,, - E,,,, = 56.5 mV is E o and the slope equals (3np1 - 3-i -a)-1. The missing parameter, using either method, is found by recourse to Fig. 3. If El,4 - E3,4 varies with disc size but this difference does not approach a limit at either large, or small, radii, then plot El,,? versus log(a). Draw a smooth curve through the points and measure its slope at each point. Next, knowing the coordinates E1,4 - E,,,, and d E,,,,/d log(a) corresponding to each voltammogram, locate a series of points on Fig. 4 and/or Fig. 5. These should all lie along a line of constant (Y,which may be thus determined. Estimate log( K o ) for each point placed on Fig. 4 or 5 and calculate k o as 40~ O/TU. Now transfer each point from Fig. 4 to Fig. 2 (and/or from Fig. 5 to Fig. 3) and read off the corresponding values of E” - El,z. Adding the experimental E,,, gives E O. varies continuously with disc size but the method of the last If El,4 - E3,4 paragraph is unsuccessful because the data lie too close to the irreversible limit, try the following alternative. Plot values of L, defined by eqn. (26), versus E,,,, - E,,, and draw the best straight line through the data from the smallest discs. Dividing 60.2 mV by the L = 0 intercept gives (II and the slope generates E o by application of eqn. (27). One now knows E o - El,, and El,., - E,,, for each voltammogram and, with (Yalso known, the problem is somewhat overdetermined when it comes to locating points on Fig. 2. E o will probably be the least accurately known parameter and allowance should be made for this in placing points at positions offering the best compromises. k o is then found in the usual way from each log( K o ). is constant for the smallest discs but becomes smaller as disc If h/4 Es/z, radius increases, first find (Y by dividing 60.2 mV by the limiting E,,, - E,,, separation. Then place points for the larger microdiscs on Fig. 2 along the known (Y contour. From each such point, read off E o - El,,? (and hence calculate E o using the experimental E,,?) and log( K ’ ) (and hence calculate k o as 40~ O/n-a ). shifts negatively with decreasing electrode size but El,., - E,,, is If El/z uniformly constant, (Ymay be found as 60.2 mV/( E1,4 - E3,4). The parameters Eo and k” are separately inaccessible, but the composite parameter (E O/25.7 mV) + ln( k o )/a is accessible via eqn. (24). CONCLUSIONS
The use of steady-state voltammetry in determining the kinetic parameters (Yand difficulties arising from and the thermodynamic parameter E o overcomes charging current and ohmic polarization interferences which often plague transient techniques. By employing a range of microdisc radii, one is able to scan a range of embracing reversible, quasi-reversible and irreversible the dimensionless K o values voltammetric behaviours. Because present fabrication techniques impose a lower
k”
96
limit of approximately 0.25 pm in microdisc radius, the applicability of steady-state voltammetry in determining kinetic parameters will be greatest for electrode reactions having standard rate constants in the range of 10e4 to IO-’ m s-l. The total kinetic spectrum may be accessed through seven windows, including the totally reversible and totally irreversible limits where most parameters are indeterminate. For the remaining five cases, (Y, k o and E o may be determined by employing distinct graphical procedures involving the quartile potentials and/or by relying on the four kinetic indicator diagrams. ACKNOWLEDGEMENTS
We express sincere thanks to our financial supporters: the Natural Sciences and Engineering Research Council of Canada and the Australian Research Grants Committee. APPENDIX
A
Here we discuss modifications which are necessary to our kinetic analyses when the diffusion coefficients Do and D, are unequal. The diffusion coefficients are related to the standard potential E o and the reuemble halfwave potential Eh=Eo-(RT/nF)ln(D,/D,)
(Al)
which follows from eqn. (9). The symbol E, is used to distinguish the reversible voltamhalfwave potential from El,?, the actual halfwave point of any steady-state mogram. Resealing of all potentials to E,, instead of E o results in the replacement of the 0 and K parameters by 1 + exp{ nF( E - E,)/RT
}
(AZ)
and
( rk “a/4D~-“D;;)
exp{ - anF( E - E,,)/RT
respectively. All our method is now assumed by Kh-
)
(A31
still apply, the only difference
being that the role of
rk”u
-
KO
(A4)
4D;-*D;;
For example, the kinetic indicator diagrams remain valid if the E o term in the ordinate is replaced by E, and the dashed lines are interpreted as corresponding to the stated values of log(Kh) rather than lo. APPENDIX
B
An expression
for quantity
nF
dE,,z
RT ln(10)
dlog(a)
nF =E
(20) may be derived G,, dln(rcO)
the first step which is a consequence
=E
by making
nF dE,/, _li dh
of the definition
d l;(;O) of KO.
use of the identity )u-,,2
97
By exploiting the relations~p [29] between the natural logarithm and the inverse h~~rbolic tangent function, the 4 = l/2 case of eqn. (19) may be rewritten 2(h+l) 3h(2h+~)/
E o ) = - 2artanh Differentiation
\
(B2)
[29] yields 12(2/P
no dE,,, --= RT dh
9h’(2h
The artanh (18) we have
function
+ 4h + V)
+ n)”
033)
4( h + 1)’
is also useful in expressing
(In f~~)~=t,~ = In in)
+ lnjl
+ 3:j~~~~j)
the logarithm
-2a
tanh(
The final term is simply anF( E,,? - E o )/RT, according the derivative of eqn. (B4) may be expressed as 2fh(2h
+ n) - (4h + a)(?2 4 l)]
dE,,,
(B3) and (B5) are inserted
F
3~1~~+I~)
From
eqn.
~
(B4)
anF dE,/, (B5)
+=dh into eqn. (Bl)
one obtains,
after
Ly+ ~3~(2~+~~-2(~+1)][3{2~~~)~+2~-4]
RT ln(l0)
nz&-G=
K*.
to eqn. (B2), and therefore
~(2~+~)[3~(2~+~)+~~~~1)] When expressions simplifications
of
i
12(2h -t nj(2h’
-t- 4h -t T)
-’ i 036)
This is the equation that was used to determine ordinate values for Figs. 4 and 5, h being calculated from eqn. (18) of the main text with q = l/2 and specified values of LYand fog( K o ). The abscissas of these figures were calculated exactly as for Figs. 2 and 3, as described in the main text. APPENDIX
C
In this appendix we consider simplifying features of quasi-reversible steady-state waves at microdisc electrodes when K o is large, though not large enough to make the voltammogram effectively reversible. In this near-reversible situation, KO is large and therefore, as eqn. (15) indicates, so will h be at all the quartile potentials. When h is large, the two bracketed terms in eqn. (18) may be expanded binomially and truncated after the first two terms
YX
leading
to 21c”
h= 3$*(1-
-fil-*)+...
(C2)
qy
For large h, eqn. (19) can be approximated
by 1
~(~,-~“)-ln~~-l)_,ni~)
3(1-q)h
which may be combined
+...
(C3)
with eqn. (C2) into (C4)
For q = l/2,
eqn. (C4) gives the simple result
-nRT nE,,,, - nE O G ___ ~FK”
showing
=
-2RTD,, cc51
Fk”a
that a plot of a!?,,,
versus l/a
will give a straight
line of slope
-2RTD,
d(G,z) d(l/a)
z
(W
Fk”
from which k” is calculable. Equation (0) shows that an extrapolation will give nE O. Experimental verification of this method has appeared elsewhere [lo]. The q = l/4 and q = 3/4 cases of eqn. (C4) lead to n(E,,4-E”)z$
(C7)
1 and
1
n(E,,,-E”)zy respectively. n( E,,, Finally, K O,
n(E,,l
We
On subtraction,
- E,,,)
= F
if we combine arrive
-E”)z
((33)
these last two results
give
m(9) + (31-a iTKl’)rRT
(C9)
eqns. (C5) and (C9) so as to eliminate
the term containing
at
(56.5 mv) - n ( Et,, - E3,4 pa_
y-1
)
(C10)
Thus, our reasoning leads to the prediction of a linear relationship, between nE,,, as Fig. 7 illustrates. The and n( E,,4 - E3,4) in the region of near-reversibility, numerical evidence in Fig. 3 confirms this predicted linearity and shows that it is most extensive for (Y values in the 0.2-0.5 range.
99
The final task in this appendix is to address the question of how large KO must be before the voltammogram is “effectively reversible”. From eqn. (C5) we see that nE, /2 is within 1 mV of its reversible value of nE” provided that -
vRT
< 1.0 mV
28.~”
which translates
(Cl11 to
>40
K”
(C12)
and explains APPENDIX
the first item in eqn. (6).
D
In this appendix we consider simplifying features of steady-state waves at microdisc electrodes when KO is small enough to make the voltammogram either totally irreversible or nearly irreversible. Equations (17) and (18) show that as K o approaches zero, which corresponds to total irreversibility, 8-l
(Dl)
and 2(h + 1)
1-q + T) -+ 4
3h(2h
The latter expression
032) shows that, if H represents
the irreversible
limit of h, then
(D3) Values of H computed from this formula for q = l/4, l/2 and 3/4 are listed in Table 5. As well, this table includes the corresponding values of ln(3rH/2), which are needed later. For future use, we also require values of the quantity defined by (D4)
TABLE
5
Parameters
associated
with the totally
irreversible
hmit
4
H
ln(3vH/2)
l/4 l/2 3/4
0.07252 0.2275 0.7545
- 1.074 0.06973 1.269
M 0.3654 1.125 3.472
To determine these, one starts with an expression and differentiates it. obtaining d(q8) p= dh
from eqn. (14)
6[(h+1)(4h+a)-h(2h+7r)] (D5)
[2(h+l)+3h(2h+7~)]~
Simplifications
M=
for q0 obtained
become
possible
when the limit is taken,
2q(H + 1) (l-q)H[3(1-q)(4H+a)-2q]
leading
to the result (D6)
-I
as the expression for M. Table 5 contains values of this parameter for q = l/4, l/2 and 3/4 calculated with the aid of formula (D6). We now turn to conditions of near-irreversibility and derive an approximate result that permits the determination of (Yand E O. When a voltammogram is close to irreversible, 0 will be slightly greater than unity and h will exceed H by an amount depending on 8. If we regard ln( h) as a function of 8. then by Taylor’s theorem lnh=(lnh)B=,+(8-1)
+ .
However, the smallness the first two, so that In h=ln
of 0 - 1 permits
(D7)
O=I
O=l
us to disregard
right-hand
terms
beyond
H+(B-l)(M+l)
(D8)
where we have incorporated the definitions (17) to replace h in eqn. (DS), we arrive at
of H and
M. If we now employ
(M+I)(@-I)-ln@=ln(K”)-ln(37rH/2)-CXln(8-1)
eqn.
(D9)
after rearrangement. Because 0 is close to unity, In 0 may be replaced by (0 - l), so that the left-hand side of eqn. (D9) becomes M(8 - 1). The expression exp{ nF( E, - E o )/RT } is next substituted for (0 - 1) in both of the terms in which it appears, leading to
=exp{g(EO--E,))[ln(KO)-ln(3aH/2)-g(E,-E”)] after each side of the equation is multiplied by exp( nF( E o - E,)/RT). Equation (DlO) is valid for any value of q. Writing it for the q = l/2 has 1.125 exp
i
gT(
E,,,
@lo) case, one
- Er,)
(Dll)
101
when numerical values are inserted from Table 5. Similarly, one may insert first the q = l/4 conditions into eqn. (DlO), and then separately set q = 3/4. Subtraction of the resulting expressions leads to 0.3654 exp
i
$
( E1,4 - Er))
- 3.472 exp{$$(E3,,4
- Er))
=exp{~(E”--E,)}[2.343--~(L.,;~--~~,~~)] lXF = RT exp {*(E~-E~)}[ RT
60.tmV
-PZ(L.,,~-E~,,~)]
The left-hand side of this equation is identical with the quantity defined eqn. (26). We see that L plotted versus n(E,,, - E,,,) gives an intercept mV)/a and a slope that conforms with eqn. (27) of the main text.
(D12) as L in of (60.2
REFERENCES 1 H. Matsuda and Y. Ayabe. Z. Elektrochem., 59 (1955) 494. 2 A.J. Bard and L.R. Faulkner, Electrochemical Methods: Fundamentals and Apphcattons, Wiley. New York, 1980. p. 220. 3 J.O. Howell and R.M. Wtghtman, Anal. Chem., 56 (1984) 524. 4 D.O. Wtpf, E.W. Kristensen, M.R. Deakm and R.M. Wtghtman, Anal. Chem.. 60 (1988) 306. 5 A.M. Bond, K.B. Oldham and C.G. Zoski, J. Electroanal. Chem.. 245 (1988) 71. 6 M. Fletschmann. S. Pow., D.R. Rohson and P.P. Schmidt (Ed%), Ultramicroelectrodes. Datatech. Morganton. NC. 1987, Ch. 3. 7 R M. Wightman and D.O. Wtpf in. A.J. Bard (Ed.). Electroanalytical Chemistry, Vol. 15. Marcel Dekker. New York. 1989, p. 268. 8 CD. Baer. N.J Stone and D.A. Sweigart, Anal. Chem.. 60 (1988) 188 9 K. Itaya. T. Abe and I. Uchida, J. Electrochem. Sot., 134 (1987) 1191. 10 K.B. Oldham. C.G. Zoski, A.M. Bond and D.A. Swetgart. J. Electroanal. Chem., 248 (1988) 467. 11 A.M. Bond, K.B. Oldham and C.G. Zoskt, Anal. Chum. Acta, 216 (1989) 177 12 C. Amatore, M.R. Deakin and R.M. Wightman, J. Electroanal. Chem., 220 (1987) 49. 13 A.M. Bond. M. Fleischmann and J. Robmson, J. Electroanal. Chem., 172 (1984) 11. 14 S. Bruckenstem, Anal. Chem., 59 (1987) 2098. 15 K.B. Oldham. J. Electroanal. Chem., 237 (1987) 303. 16 K.B. Oldham. J. Electroanal. Chem.. 250 (1988) 1. 17 A.M. Bond, M. Fleischmann and J. Robmson, J. Electroanal. Chem., 168 (1984) 299. 18 M. Ciszkowska and Z. StoJek, J. Electroanal. Chem., 213 (1986) 189. 19 H.A. Laitinen and I.M. Kolthoff. J. Phys. Chem., 45 (1941) 1062. 20 R.M. Wightman, Anal. Chem.. 53 (1981) 1125A. 21 K. Aokt, K. Akimoto, K. Tokuda, H. Matsuda and J. Osteryoung, J. Electroanal. Chem., 171 (1984) 219. 22 C.G. Zoski, A.M. Bond, C.L. Colyer. J.C. Myland and K.B Oldham. J. Electroanal. Chem., 263 (1989) 1. 23 Y. Saito, Rev. Polarogr. (Jpn), 15 (1968) 177. 24 J. Tomes, Collect. Czech. Chem. Commun., 9 (1937) 150. 25 K.B. Oldham and C.G. Zoski, J. Electroanal. Chem.. 256 (1988) 11. 26 A.C. Mtchael. R.M. Wightman and C.A. Amatore, J. Electroanal. Chem., 263 (1989) 163. 27 Z. Galus. J. Golas and J. Osteryoung. J. Phys. Chem., 92 (1988) 1103. 28 K. Aoki, K. Tokuda and H. Matsuda. J. Electroanal. Chem.. 235 (1987) 87. 29 J. Spamer and K.B. Oldham, An Atlas of Functions, Hemtsphere, New York and Springer, Berlin. 1987, Ch. 31.
J. Electroanal. Chem., 270 (1989) 79-101 Elsevier Sequoia S.A.. Lausanne - Printed
in The Netherlands
Kinetic parameters from steady-state microdisc electrodes
voltammograms
at
Keith B. Oldham and Janice C. Myland Department
of Chemrstr).
Trent Unroersrt.v, Peterborough,
Ontarro. K9J 7B8 (Cunudu)
Cynthia G. Zoski and Alan M. Bond Depurtment (Received
of Chemrcai and Ana!vtrcul Scrences. Dealrm Unwer.rrty. Geelong, Vwtorru 3217 (Austraba) 21 March
1989)
ABSTRACT Exact methods of determinmg the kinetic parameters a and k”, as well as the thermodynamic parameter E O, from steady-state voltammograms at mIcrodIsc electrodes are presented. These methods rely on employing a range of mlcrodw radii in order to scan the reversible, quasi-reversible or irreversible kmetic regimes. The experlmentally determined quartlle potentials E, ,4, E,, ?. and E, ,4 are either graphed or used to locate pomts on the so-called “kmetic Indicator diagrams” from which a. k O. and E” may be determined. Our procedures are not based on the usual assumptions of “uniform accessibility” or equal diffusion coeffu.zients.
INTRODUCTION
Two parameters, the transfer coefficient (Y and the standard heterogeneous rate constant X-O, are used to characterize the kinetics of simple electron transfer reactions such as the reduction O(soln)
+ n eP -j R(soln)
(1)
the In addition, the standard potential E o must also be known to characterize reaction fully. Voltammetric experiments may be classified according to whether the electron or “irreversibly”. This classifitransfer step occurs “reversibly”, “quasi-reversibly”, cation of a particular experiment determines which kinetic parameters, if any. are accessible, as detailed in Table 1. The basis of the classification, for conventional transient voltammetric experiments, involves a comparison of the ratio Do/(kO)’ with some characteristic time 7 [l]. Here Do is the diffusion coefficient of the electroreactant and, depending on the nature of the experiment, r may be a drop 0022-0728/89/$03.50
‘is 1989 Elsevier Sequoia
S.A.
80
TABLE
1
Classification
of voltammetrlc
regimes
and the kinetic parameters
which are experimentally
accessible
Regime
n
ho
reversible quasi-reversible irreversible
maccessible accessible accewble
maccesslble accessible accessible accessible maccesslble. though a parameter involving both h-O and E 0 is calculable
time, a pulse width, a reciprocal sweep rate u. One has Do/( k o )” cc
reversible
Do/(/?“)‘=
quasi-reversible
7:
Do/( k o )’ > 7:
frequency,
E”
or the quantity
RT/nFu
involving
a
(2)
irreversible
As is evident from Table 1, experiments in the quasi-reversible regime are the most advantageous for kinetic studies. Because diffusion coefficients have magnitudes close to lo-9 m’ s-l and many interesting electrode reactions have standard rate constants of lo-” m s-l or greater, we see that characteristic times of one millisecond or less are required to attain quasi-reversibility. There are, however, a number of experimental difficulties with transient voltammetry on such timescales, notably interference from the large charging currents that flow under such conditions [2-41. A way around these difficulties is to espouse steady-state experiments. In the steady state there is no charging current whatsoever, and no characteristic time either. Instead, the role of determining the voltammetric regime is taken over by a characteristic length. X. dependent upon the dimensions of the electrode. It is the comparative magnitude of this length and the quotient Do/k” that is crucial [5]. One has Do/k”
<
reversible
Do/k”
= A:
quasi-reversible
Do/k”
>> A:
irreversible
(3)
For Do = 10e9 rn’ s-’ and k” = 10-j m SC’, one needs a characteristic electrode dimension h of order one micrometre to achieve quasi-reversibility. Recently there has been much activity and success in the fabrication of electrodes of micrometre and submicrometre dimensions [6-91 opening the door to the use of steady-state voltammetry [lO,ll] as a means of measuring electrochemical kinetic parameters. Perhaps the most easily constructed microelectrodes are those with the geometry of an inlaid disc [6-91, in which a flat circular electronically conducting disc is embedded in an insulator whose surface forms a geometric continuation of the electrode/solution interface. It is with such microelectrodes that this article is solely
81
concerned. For inlaid disc microelectrodes we may associate the characteristic length, introduced in the last paragraph, with the term 7ru/4 where a is the radius of the microdisc. Further, we define a dimensionless parameter K* by K0
=
hk o,‘D, = nk “a/(4Do)
(4)
so that in the light of eqn. (3) one has It”>l:
reversible quasi-reversible
K o 21: K”
(<
(5)
irreversible
1:
In theory, full reversibility and total irreversibility are conditions that are approached as limits, so that all experiments may be regarded as quasi-reversible. In practice, however, the useful range of quasi-reversibility embraces only about two orders of magnitude of the KO parameter, as shown by the classification K”
>
0.2 5 K”
<
40: K* 0.1:
effectively <
20:
usefully
reversible
quasi-reversible
effectively
(6)
irreversible
The purpose of this article is to propose practical and convenient methods by which the parameters LY,k *, and E o may be obtained from steady-state voltammograms recorded using inlaid disc microelectrodes of a variety of radii. Our methods are able to handle voltammetric data obtained from experiments conducted under quasi-reversible conditions, sometimes augmented by purely reversible or totally irreversible data. Only a limited range of k“ values is accessible. Table 2, which is based on the typical diffusion coefficient value Do = 1 X 10e9 m2 s-‘, lists the quasi-resmallest and largest disc radii that cause KO to fall within the “usefully versible” range specified in eqn. (6). The present state of the art of microdisc fabrication limits the use of discs smaller than about 0.25 pm in radius [9]. Large discs are easily made but the larger the disc, the longer it takes to achieve a steady state, and currents at potentiostatted discs of radii much larger than 25 pm take so long to reach constancy that convection often intervenes [ll]. Accordingly, the present methods will find greatest applicability in studying electrode reactions having standard rate constants in the range 10 m4 to lo-’ m SC’.
TABLE
2
The smallest k”/m lo-’ 10-z 10-j 10-4 10-5
s-’
and largest
disc radii that cause k * to fall wthin
the “usefully
a/pm 0.00250.25 0.015 2.5 0.25 25 2.5 - 250 25 -2500
quasi-reversible”
range
x2
In this work it is assumed that diffusion is the sole transport mode, conditions having been selected to obviate convection and migration. Of course, forced convection can be avoided by not agitating the solution but, as mentioned earlier, natural convection may obtrude unless the disc electrode is small enough that a steady state is reached sufficiently rapidly. To prevent migration, supporting electrolyte will have been added to the solution phase. A serendipitous feature of voltall~metry at microelectrodes is that for certain electrode reactions (those that increase the ionic strength at the interface) far less supporting electrolyte is needed than is the case for studies with macroelectrodes [I?-181. VOLTAMMETRIC
SHAPES
Steady-state voltammograms may be recorded in a point-by-point fashion by performing a series of chronoamperometric experiments, each at a slightly different potential [19]. More conveniently, they result from applying a potential ramp to the microdisc electrode, provided that the sweep rate is slow enough that the voltammogram retraces itself when the direction of the ramp is reversed [3,8.20.21] (though a means of circumventing this difficulty has recently been suggested 1221). A third method would be to record the ultimate potentials reached during a series of chronopotelltiometric experiments each at a slightly different current (not exceeding the limiting current), while a fourth could employ a slowly increasing current ramp. It is a valuable characteristic of steady-state voltammetry that the resulting current-voltage curves are totally independent of the details of the method used to obtain them [tl]. Figure 1 shows the typical shapes of reversible, quasi-reversible and irreversible steady-state voltammograms that result from experiments at microdisc electrodes. Note that the same limiting current 15,231 d, = 4~~Do~~U
(7)
is attained irrespective of the degree of reversibility, but that the slope of the voltammetric wave diminishes as one proceeds from reversibility through quasi-reversibility to irreversibility. The three curves in Fig. 1 correspond to reductions of Do) three substances (all of equal bulk concentrations c:, and diffusion coefficients that share the same values of a! and E* but differ in their standard rate constants. It is evident from the figure that the waves not only become less steep but also shift towards more negative potentials as k” decreases. Ret~ersible waves The shape of a reversible
steady-state
voltammogram
obeys the simple
equation
PI i=i,/fl+(Do/L),)exp[njc(E:-E’)/RT]]
(8)
83
0.50-
0.25-
I
I
I
200
I
-200
0
I
-4c
Fig. 1. The shapes of reversible (k o = co), quasi-reversible (k o = 1.6 Do/au) and effectively irreversible (k” = 0.4 Do/m) steady-state voltammograms. The parameter values (Y= 0.5, n =l.O, Do/D, =l.O and T = 298.2 K were assumed.
relating current i to potential E. From this it follows that the difference the standard and half-wave potentials satisfies the relationship n(E”
-E1,2)
= CRT/F)
ln(D,/D.)
between
(9)
Because the diffusion coefficient D, of the product of the electrode reaction rarely differs significantly from that of the reactant, the half-wave and standard potentials are virtually identical for reversible electrode reactions. A useful way of quantifying the slope of a voltammetric wave was pioneered by Tomes [24] and involves measuring the difference between the one-quarter-wave potential and the three-quarters-wave potential. From eqn. (8) one can show that, for a reversible n-electron reduction wave, this potential difference is given by n ( E,,4 - E3,4) = (RT/F)
ln(9) = 56.5 mV
(10)
and is not influenced by any disparity between Do and D,. Notice that, as presaged in Table 1, no information about (Yor k” is contained in eqn. (9) or (10). The millivoltage given in eqn. (lo), and in every potential difference cited subsequently in this article, relates to 25.0 o C. For other temperatures T, the stated millivoltages (including those marked on the axes of diagrams) should be multiplied by T/(298.2 K). Quasi-reversible
waves
The shape of quasi-reversible steady-state voltammograms from microdisc electrodes has been the main subject of several publications from these laboratories
84
[5,11,25]. In terms of a parameter
8, defined
by
8=1+(D,/D,)exp[nF(E-E”)/RT] and a second parameter
(11)
K,
defined
by
K=K”exp[-OLnF(E-Eo)/RT] the current-potential
(12)
relationship
for a quasi-reversible
in recent work by Michael as
This equation has been verified independently Equation (13) can be written more succinctly 1 --I= 40
wave is given by [25]
et al. [26].
2(h + 1) 3h(2h
(14)
+a)
where k is a new parameter, ,=%=%[I+2
proportional
to the
exp{$(E-E”)i]
while q is simply
K$
product,
and defined
by (75)
exp(s(E-E”))
the height of the voltammetric
current
q = i/i,
relative
to the plateau (16)
Analogues of eqns. (9) and (10) do not exist for quasi-reversible conditions and one must be content with numerical or graphical equivalents. In experimental practice, one rarely has accurate data on the diffusion coefficients of both 0 and R, so it is customary to assume Do = D,. We shall follow this custom by henceforth ignoring the (D&D,) multipliers in eqns. (11) and (15). With this simplification, these two equations may be combined into 21c08=3rk(d-l)a
(17)
In Appendix A, we discuss the minor modifications that are necessary if the equality of Do and D, is not assumed. Our interest focuses on the three quartile potentials: E,,4, E,,z and EX,4, which we represent collectively by E,, with q = l/4, l/2 or 3/4. For given values of q, a the voltage n( E, - E “) by first using an iterative and KO, one can determine method to solve the equation 2(k+l) 3k (2k + T)
Ii
for k and then substituting $(E,-EO)=Lni[q+
a 1+
2(k+l) 3k(2k
+ T) I
'-a_
2K”
3n.q
(18)
this value into ~(2hh=1n))]-~-1)
(19)
85
Equation (18) is derived by eliminating 6 between eqns. (14) and (17). while eqn. (19) follows directly from eqn. (14) and the definition of 8. We employed the procedure of the last paragraph to calculate values of n( E O E,,,), n(E” - E,,4) and n( E o - E,,z) at round values of a: and log. Subtraction of n(E” -I?,,,) from n(E’ - E,,,) then generates the potential difference - E3,4) separating the one-quarter-wave and three-quarters-wave potentials. n(E,,, Because values of n(E * - E,,?) are also known at the same round values of (Yand log{K” ), Figs. 2 and 3 could be constructed. These diagrams are small-scale and large-scale versions of plots of n( E o - E,,,) versus n(E,,, - E,,,,) in which “contour” lines link points of equal (Yand points of equal logfrr 0 ). They show how the slopes [as represented by n( E,,4 - E3,4)) and positions [as represented by n(E” - E,,2)] of quasi-reversible steady-state waves at microdisc electrodes depend on cf and log( Ko )_ To locate a point on either Fig. 2 or 3 requires that one know a value for the standard potential E o in order to calculate the ordinate n( E o - E,,z). Unforvoltammograms tunately, the determination of E” from a series of steady-state recorded at microdiscs of various radii may not always be possible. Thus, it may be convenient (and experimentally is always possible) to measure instead the change in half-wave potential E,,,2 with disc radius a and thereby calculate one or more values of the n
dE,,,/d
log a
(20)
quantity. These experimental values may then be used in conjunction with Figs. 4 and/or 5, which show how the~quantity (20) is related theoretically to the Tomes cx and log(~ o ). Appendix B separation n (E,,, - E3,4) and the kinetic parameters describes the construction of Figs. 4 and 5. These diagrams are small-scale and large-scale versions of plots of nd E,,,/d log a versus nf E,,, - E3,_,) in which contour lines again link points of equal (Yand log( K o ) as in Figs. 2 and 3. Figures 4 and 5 show how the effect of the disc radius on the half-wave potential [as represented by dE,,,/d log a) and the wave slope [as represented by n( E,,,, - xT;,~)] of quasi-reversible steady-state voltammograms depend on (Y and log(K”). We call Figs. 2-5 “kinetic indicator diagrams”. Irreversible
waves
Total irreversibility corresponds to the reduction wave being displaced negatively from the standard potential to such an extent that, at all points on the escarpment of the voltammogram, the term exp[nF( E - E * )/RT] is negligible in comparison with unity. It follows from eqn. (11) that 8 is effectively equal to one. Hence eqn. (13) reduces to the simple result
(21) for an irreversible
wave.
86
v.
7
,\$ /-
1’ . ./’. . / .
,
. 1-Y.
.o\
/:
.
.
*
.
/
.
..’ /
+
/
’ /: c
;/
_.+y
. . . __./
.,I-
. . .
.
.A..
/
-.
/_,‘.
.
.
/Q~K
fi
. #
c
c_
. a= 60
80
i
.oo
5z&,mV I
20
_w+
,,,A- 0 2. . . .. 4: Z0~T.z-L _ A__-____;_
I i
1_
114
140
160
f
5/4
I
180
2013
Fig. 2. A small-scale kinetic indicator diagram which shows the relationshlp between n( E o - E,,z) n (E, ,4 - E,,,) at constant a or log( K o ) for steady-state voltammograms at microdlsc electrodes.
Setting i/i,
equal to one half in eqn. (21), one finds
n(E" - E,,z)= (RT/c&)[0.06973
- ln(ic’=)]
K =
and
1.072 so that (22)
when eqn. (12) is invoked. Similar expressions involving E3,4and E,,4emerge from setting i/i, equal to 3/4 or l/4 respectively in eqn. (21). Subtraction of these two latter expressions leads to [25] n(
E,,4- E3,4)= 2.342RT/aF= (60.2/a)mV
(23)
87
4c
35
3c
25
20
15
10
,0-g . . . . .
S
9_
I
60
Fig. 3. The expanded
65
70
75
inset of Fig. 2.
Hence, as reported in Table 1, a value of (Y is easily accessible from an irreversible steady-state voltammetric wave at an inlaid disc microelectrode simply by measuring the separation of the one-quarter-wave and three-quarters-wave potentials. For an irreversible wave, this separation is dependent on neither the disc radius a nor the rate constant k”. Thus, experimental evidence for an irreversible wave is a constant Tomes potential difference as the disc is diminished in size. Returning to eqn. (22), one finds 8.0 mV + 25.7 mV ‘F ln(k”) ln(Do/a)=nE”+(24) n-G/, + ~ (Y Ly
88 >:
16tI- .<
:
:
:
:
1.
“:....:““:....:..
n
.
.‘(5
:
14tI- ._
I&$(?$?7 :F : /
. . . . . . . . . . . . . . . . . .
F
;
:
.
F
12cI\
*
*
Y) h.
L 0-
1OC I-
.,r,!.
./r..
./.
. .
.,.-r-i.
8C
6C 9
4c
I<
_--. . . . . . .
Ly-T-7-y . . . . . . . . . . .._..............
20 IFig 0
J
‘Ei0_
Fig.
P
4.
d&,/d microdisc
A
1
1
I
I
I
80
100
120
5
small-scale
kinetic
indicator
diagram
1 ’ . n CE1/4-E3;41 /III\ J I
I
140 whxh
160 portrays
the
I
180 relationship
1
200 between
n
E,,,) at constant a or log( I(’ ) for steady-state voltammograms at log(a) and n(E,,,electrodes. The unlabelled log(K “) lines correspond to -0.4, - 0.6, - 0.8, - 1.0 and - co.
which explains the allusion in Table 1 to the accessibility from an irreversible experiment of a parameter involving both k ’ and E ‘, though neither of these quantities is separately accessible. METHODS
OF DETERMINING
KINETIC
PARAMETERS
The principle of using steady-state voltammograms at microdisc electrodes to measure the kinetic parameters, LYand k O, as well as the thermodynamic parameter
4c
35
30
2s
20
15
10
5
0 c Fig. 5. The expanded
inset of Fig. 4.
E O, of reaction (1) relies on employing a range of disc radii u. In this way, one is able to scan a range of values of the dimensionless parameter ICO (because this equals vrk”a/4Do). We have already remarked that the usable range of disc radii is rather limited and a similar limitation consequently afflicts K”. As portrayed in classification (6), the total spectrum of KO values embraces reversible, quasi-reversible and irreversible voltammetric behaviours. The small accessible range of KO may intersect this total spectrum in a variety of ways, as illustrated in Fig. 6. If the accessible range of KO values lies entirely within the reversible region, as illustrated by Case 0 in Fig. 6, or entirely within the irreversible region, as illustrated
90
Case VI ,*YF*YT*Y/-. y///‘.“““‘y/A / Irreversible \
I
-3
I
-2
Case II
Case IV
Cose 0
v”“““““/g v/,““““y///// /Quasireversible Reversible ,\ / 4 Case V Case III Case I
I
-1
I
II
I 1
I
2
log (K”)
I
>
3
Fig. 6. The “spectrum” of voltammetric expenments, showmg how the Irreversible, quasi-reversible and reversible kmetlc regimes reflect log( Ko ). The seven cases dlagrammed are discussed m turn in the text.
by Case VI, then only one parameter is determinate. However, for each of the other five cases diagrammed in Fig. 6, the experimentally determined quartile potentials (Y, k o (Q/47 El/z. and E,,,) may be employed to determine the three parameters must be employed in each case. All these and E”, though a distinct procedure procedures, which will now be discussed in turn, make use of one or several of the kinetic indicator diagrams, Figs. 2-5. Other approaches to this problem have been considered recently by Galus et al. [27] and by Aoki et al. [28].
Method I: reversible/near-reversible
Case I in Fig. 6 shows a situation in which the accessible range of KO includes portions of both the reversible and quasi-reversible regimes. One can identify this situation from the fact that the largest disc will show a n( E,,, - E,,,,) separation of 56.5 mV within experimental error, as predicted by eqn. (lo), whereas smaller discs will generate voltammograms whose one-quarter and three-quarters potentials are separated further. In this case, the half-wave potential developed by the largest disc will obey eqn. (9) and will therefore equal the standard potential E o (recall that we are assuming that the two diffusion coefficients are equal: see Appendix A if this assumption is rz( E, ,,4 - E3,4) not made). Now knowing E O, one may calculate the coordinates and n(E” - E,,2) of a point on kinetic indicator diagram Fig. 3 for the voltammogram from each of the smaller discs. The location of such an experimental point on the kinetic indicator diagram permits the simultaneous estimation of (Y and of log( K o ). Thence, via the relationship k” = 4D,K”/Ta
(25)
the standard rate constant k” may be determined. Each experimental voltammogram from a smaller disc yields a distinct point on the kinetic indicator diagram. The points should lie on, or parallel to, lines of constant cll, but will correspond to various values of log( K”) because this parameter incorporates the variable disc radius. Nevertheless, use of eqn. (25) should lead to a unique k o value.
91
Method II: near-reversible/quasi-reversible As illustrated in Fig. 6, Case II corresponds to a situation in which the accessible values of KO do not enter the reversible region, but lie close to it. That one has encountered this situation will be evident from the fact that the experimental Tomes differences, n ( E1,4 - E3,4), will all exceed 56.5 mV but will approach this value for the largest discs. The procedure that we advocate in this case involves an extrapolation into the reversible region, yielding a value of the standard potential E O. In Appendix C (and elsewhere [lo]) we demonstrate that, in this near-reversible regime. a graph of nE,,, versus l/a will yield a straight line of slope - 2RTD,,/Fk” and intercept nE O, thus providing a very convenient method for determining both k o and E O. Alternatively, in the region of near-reversibility a linear relationship exists between n ( El,z -E”) and n(E,,, - E3,4) for discs of differing radii, as predicted by eqn. (ClO) and illustrated in Fig. 7. The extrapolation indicated in Fig. 7 is used to find EO, while the slope l/(3”-’ - 3’-,) of the line can be used to provide a value of (Y. The values for the eight data pairs were calculated as described previously and are reported in Table 3 along with the corresponding disc radii. The line in Fig. 7 has been drawn in accordance with eqn. (ClO). The points corresponding to near-reversibility lie on, or close to, the line whereas the less reversible points show a mild departure from linearity. Nevertheless, a regression line drawn through all eight points would give a value of (Y(calculated from the slope) which is in error by only 3%. An alternative method of determining (Y is from the kinetic indicator diagram Fig. 3. Knowing E o and k” (and hence KO ) from a graph of nE,,, vs. l/a, one
nE,,JmV
-llO-
I '
56
I 58
I 60
I 62
nK,,4-E,,,) /mV I 64
I 66
Fig. 7. Graph of half-wave potential versus the difference between three-quarters-wave potential for the hypothetical electroreductlon
the one-quarter-wave potential detailed m Table 3.
and the
92 TABLE
3
Calculated halfwave potenttals and Tomes potential differences E”=-lOO.OmV, k”=O.Olms-‘, D,,=D,=1.0X10-9m’ microdisc electrodes of the stated radit. Values of nE,,, and difference was calculated are also tabulated
for a hypothettcal electroreductton with sm’,T=298.2K, n=land oc=0.5,at nE,,, from whrch the Tomes potenttal
a/pm
nE,,jZ/mV
nEt,,/mV
nE:,.,/mV
n(E, ,4 - .%,)/mV
0.51 0.64 0.80 1.01 1.21 2.02 3.20 8.03
-
-
-
66.4 64.5 63.0 61.7 60.7 59.2 58.2 57.2
108.3 106.8 105.5 104.5 103.6 102.4 101.5 100.6
16.6 15.1 75.0 14.4 73.9 13.2 72.7 72.1
143.0 140.2 138.0 136.1 134.6 132.4 130.9 129.3
can easily locate a series of points along one of the (Ycontour lines, thus providing a measure of (Y.The value of (Yso obtained may be compared to that calculated from the slope of a plot resembling Fig. 7. Method III: quasi-reversible In Case III, experimentally accessible values of ICO lie solely in the quasi-reversible regime. Experimental confirmation that one has encountered this situation will be reflected in the n( E,,, - E,,,) potential separations, which will not approach a limit at either the smallest or the largest discs studied. In this case, one must rely solely on the kinetic indicator diagrams. Since E o will generally not be known, one resorts first to Figs. 4 and 5 which make use of the shift of n&,2 with electrode size in relation to n( E,,4 - E,,,,). This permits values of (Y simultaneously. Values and log( K o ) [and hence k o via eqn. (25)] to be determined of OLand log(~O ) in addition to n( E,,, - E,,,,) provide the necessary information for locating a point for each voltammogram on the Fig. 2 or 3 kinetic indicator diagram which relates n( E o - E,,, ) and n( E,,4 - E,,,, ). From a particular value a unique E o value may be found. of n(E” - E,,z) for each voltammogram, Method IV: near-irreversible As portrayed in Fig. 6, Case IV corresponds to a situation in which the accessible values of KO do not enter the irreversible regime, but lie close to it. One can identify which will increase this situation experimentally by the n( E,,4 - E,,.,) separations with decreasing disc size and approach a limit [the 60.2 mV/a irreversible limit] with the smallest discs. The procedure that we advocate in this case involves an extrapolation into the of each voltammogram irreversible regime. Measure the E,,4 and E,,, potentials and plot the quantity L = 0.3654 exp
n(El14- E,) 25.1 mV
- 3.472 exp
n(Q4 - -%I\ 25.7 mV
1
93
642-
Fig. 8. Graph of the experimentally calculated parameter L versus the measured one-quarter-wave three-quarters-wave potential separation for the hypothetical electroreduction reported m Table 3.
and
versus n( E,,4 - E,,,). Here E, is any constant reference potential in the vicinity of the voltammetric waves; it could, for instance, be the half-wave potential of one of the voltammograms. In Appendix D, we demonstrate that. in this near-irreversible linear and may be extrapolated to regime, such a plot will be approximately intersect the n( E,,4 - E,,,) axis at (60.2 mV)/u. This intercept permits a determination of a. while the standard potential is then calculable from the slope by using the result (25.7 mV) ln{ -(25.7 mV)(slope)/a} tlE” =nE,+ (27) To illustrate this procedure, we have prepared Fig. 8. Table 4 tabulates the values for the eight data points at the corresponding disc radii. The solid line of Fig. 8 was
TABLE
4
Data for the Fig. 8 graph of the experimentally determined parameter L (defined in eqn. 26) versus the one-quarter-wave and three-quarters-wave potential separation. Values of k o = 1 x 10e4 m s- ‘. D = 1.0 x 10m9 rn’ SC’. n = 1. a = 0.5. T = 298.2 K, E o = - 100.0 mV and E, = - 200.0 mV were assumed. The numerical value of the disc radii listed in the first column led to the remainmg four columns of data
a/v
L
fi (6
0.80 1.3 1.6 2.0 2.5 3.2 4.0 5.1
0.5422 1.310 2.003 3.009 4.411 6.260 8.537 11.12
119.8 119.0 118.2 117.1 115.5 113.2 110.3 106.6
,4
-
4,
)/mV
ME,,., -- E,)/mV 12.6 35.3 46.3 56.9 66.9 16.2 X4.6 92.0
n(E3, 4 - 107.2 - 83.6 -11.9 - 60.2 -486 - 37.0 - 25.7 - 14.6
-
E,)/mV
94
drawn according to eqn. (D12). The points corresponding to near-irreversibility lie on, or close to, the line while the more reversible points depart from the line of theoretical slope. A regression line drawn through all eight points leads to a standard potential which is in error by 4 mV and a value for the transfer coefficient which is 0.3% low. Now, knowing E o and (Y,one may calculate the coordinates n( E o - E,,,,) and - E3,4) of a point on kinetic indicator diagram Fig. 2 for the voltammogram n(E,,, from each of the discs. The location of each experimental point on the kinetic indicator diagram permits the determination of a unique value of log( K o ) and hence k o from relationship (25). METHOD
V: NEAR-IRREVERSIBLE/IRREVERSIBLE
In this last case, the accessible range of K a values includes portions of both the quasireversible and irreversible kinetic regimes. One can identify this situation experimentally from the fact that the smallest discs will have identical n ( E, ,4 - E3,4) separations, whereas larger discs will generate voltammograms whose one-quarter and three-quarters potentials are less separated. For the smallest discs, the n( E,,4 - E?,,) separation will obey eqn. (23), permitting cx to be calculated as (60.2 mV)/(nE,,, - nE,,,). One can then locate the appropriate (Ycontour on Fig. 2 and insert points corresponding to the larger discs at the intersections of this contour with the measured n( E,,, - E,,,,) abscissas. From each such point, values of n (E o - E,,,,) and log( KO) can be read off. From the first. E o is calculable using the measured E,,,, while from the second, k o may be found via eqn. (25). SUMMARY
OF METHODS
Here we summarize the methods advocated for the determination of the parameters E”, k o and cx from measured steady-state values of E,,4, E,,?, and E,,, obtained from microdisc voltammograms using a series of disc radii u. In this summary we make the simplifying assumptions that n = 1. T = 298.2 K and that the electroreactant has a known diffusion coefficient D equal to that of its reduction product. If no variation with disc radius, beyond what can be attributed to experimental scatter, is observed in El,?. and E,,4 - E,,, - 56.5 mV, then (Y is indeterminate, E o can be equated to an average E,,,, and k o exceeds 51 D/a m,n where a m,n is the radius of the smallest electrode. If E,,, - E3,4 equals 56.5 mV for the largest discs but exceeds this value for the smaller electrodes, E o may be equated to an average E,,2 for the largest electrodes. For each of the smaller electrodes, locate the point on Fig. 3 that corresponds to the coordinates E, ,4 - E,,, and E o - E,,?. All these points should lie along a line of constant CX,whence this parameter may be found. Estimate log( K o ) for each point and calculate ~DK~/~u; a constant value, equal to k O, should be obtained.
95
If all values of E,,4 - E3,4 exceed 56.5 mV but this value is approached as the disc size increases, then two methods are available. Both require that E o be found by extrapolation. In the first method, plot E,,,, versus l/a: the intercept at I/a = 0 is E” and k” is equal to (-51.4 mV)D/(slope). In the second method, plot E,,z versus E,,4 - E, 4; the intercept at E,,, - E,,,, = 56.5 mV is E o and the slope equals (3np1 - 3-i -a)-1. The missing parameter, using either method, is found by recourse to Fig. 3. If El,4 - E3,4 varies with disc size but this difference does not approach a limit at either large, or small, radii, then plot El,,? versus log(a). Draw a smooth curve through the points and measure its slope at each point. Next, knowing the coordinates E1,4 - E,,,, and d E,,,,/d log(a) corresponding to each voltammogram, locate a series of points on Fig. 4 and/or Fig. 5. These should all lie along a line of constant (Y,which may be thus determined. Estimate log( K o ) for each point placed on Fig. 4 or 5 and calculate k o as 40~ O/TU. Now transfer each point from Fig. 4 to Fig. 2 (and/or from Fig. 5 to Fig. 3) and read off the corresponding values of E” - El,z. Adding the experimental E,,, gives E O. varies continuously with disc size but the method of the last If El,4 - E3,4 paragraph is unsuccessful because the data lie too close to the irreversible limit, try the following alternative. Plot values of L, defined by eqn. (26), versus E,,,, - E,,, and draw the best straight line through the data from the smallest discs. Dividing 60.2 mV by the L = 0 intercept gives (II and the slope generates E o by application of eqn. (27). One now knows E o - El,, and El,., - E,,, for each voltammogram and, with (Yalso known, the problem is somewhat overdetermined when it comes to locating points on Fig. 2. E o will probably be the least accurately known parameter and allowance should be made for this in placing points at positions offering the best compromises. k o is then found in the usual way from each log( K o ). is constant for the smallest discs but becomes smaller as disc If h/4 Es/z, radius increases, first find (Y by dividing 60.2 mV by the limiting E,,, - E,,, separation. Then place points for the larger microdiscs on Fig. 2 along the known (Y contour. From each such point, read off E o - El,,? (and hence calculate E o using the experimental E,,?) and log( K ’ ) (and hence calculate k o as 40~ O/n-a ). shifts negatively with decreasing electrode size but El,., - E,,, is If El/z uniformly constant, (Ymay be found as 60.2 mV/( E1,4 - E3,4). The parameters Eo and k” are separately inaccessible, but the composite parameter (E O/25.7 mV) + ln( k o )/a is accessible via eqn. (24). CONCLUSIONS
The use of steady-state voltammetry in determining the kinetic parameters (Yand difficulties arising from and the thermodynamic parameter E o overcomes charging current and ohmic polarization interferences which often plague transient techniques. By employing a range of microdisc radii, one is able to scan a range of embracing reversible, quasi-reversible and irreversible the dimensionless K o values voltammetric behaviours. Because present fabrication techniques impose a lower
k”
96
limit of approximately 0.25 pm in microdisc radius, the applicability of steady-state voltammetry in determining kinetic parameters will be greatest for electrode reactions having standard rate constants in the range of 10e4 to IO-’ m s-l. The total kinetic spectrum may be accessed through seven windows, including the totally reversible and totally irreversible limits where most parameters are indeterminate. For the remaining five cases, (Y, k o and E o may be determined by employing distinct graphical procedures involving the quartile potentials and/or by relying on the four kinetic indicator diagrams. ACKNOWLEDGEMENTS
We express sincere thanks to our financial supporters: the Natural Sciences and Engineering Research Council of Canada and the Australian Research Grants Committee. APPENDIX
A
Here we discuss modifications which are necessary to our kinetic analyses when the diffusion coefficients Do and D, are unequal. The diffusion coefficients are related to the standard potential E o and the reuemble halfwave potential Eh=Eo-(RT/nF)ln(D,/D,)
(Al)
which follows from eqn. (9). The symbol E, is used to distinguish the reversible voltamhalfwave potential from El,?, the actual halfwave point of any steady-state mogram. Resealing of all potentials to E,, instead of E o results in the replacement of the 0 and K parameters by 1 + exp{ nF( E - E,)/RT
}
(AZ)
and
( rk “a/4D~-“D;;)
exp{ - anF( E - E,,)/RT
respectively. All our method is now assumed by Kh-
)
(A31
still apply, the only difference
being that the role of
rk”u
-
KO
(A4)
4D;-*D;;
For example, the kinetic indicator diagrams remain valid if the E o term in the ordinate is replaced by E, and the dashed lines are interpreted as corresponding to the stated values of log(Kh) rather than lo. APPENDIX
B
An expression
for quantity
nF
dE,,z
RT ln(10)
dlog(a)
nF =E
(20) may be derived G,, dln(rcO)
the first step which is a consequence
=E
by making
nF dE,/, _li dh
of the definition
d l;(;O) of KO.
use of the identity )u-,,2
97
By exploiting the relations~p [29] between the natural logarithm and the inverse h~~rbolic tangent function, the 4 = l/2 case of eqn. (19) may be rewritten 2(h+l) 3h(2h+~)/
E o ) = - 2artanh Differentiation
\
(B2)
[29] yields 12(2/P
no dE,,, --= RT dh
9h’(2h
The artanh (18) we have
function
+ 4h + V)
+ n)”
033)
4( h + 1)’
is also useful in expressing
(In f~~)~=t,~ = In in)
+ lnjl
+ 3:j~~~~j)
the logarithm
-2a
tanh(
The final term is simply anF( E,,? - E o )/RT, according the derivative of eqn. (B4) may be expressed as 2fh(2h
+ n) - (4h + a)(?2 4 l)]
dE,,,
(B3) and (B5) are inserted
F
3~1~~+I~)
From
eqn.
~
(B4)
anF dE,/, (B5)
+=dh into eqn. (Bl)
one obtains,
after
Ly+ ~3~(2~+~~-2(~+1)][3{2~~~)~+2~-4]
RT ln(l0)
nz&-G=
K*.
to eqn. (B2), and therefore
~(2~+~)[3~(2~+~)+~~~~1)] When expressions simplifications
of
i
12(2h -t nj(2h’
-t- 4h -t T)
-’ i 036)
This is the equation that was used to determine ordinate values for Figs. 4 and 5, h being calculated from eqn. (18) of the main text with q = l/2 and specified values of LYand fog( K o ). The abscissas of these figures were calculated exactly as for Figs. 2 and 3, as described in the main text. APPENDIX
C
In this appendix we consider simplifying features of quasi-reversible steady-state waves at microdisc electrodes when K o is large, though not large enough to make the voltammogram effectively reversible. In this near-reversible situation, KO is large and therefore, as eqn. (15) indicates, so will h be at all the quartile potentials. When h is large, the two bracketed terms in eqn. (18) may be expanded binomially and truncated after the first two terms
YX
leading
to 21c”
h= 3$*(1-
-fil-*)+...
(C2)
qy
For large h, eqn. (19) can be approximated
by 1
~(~,-~“)-ln~~-l)_,ni~)
3(1-q)h
which may be combined
+...
(C3)
with eqn. (C2) into (C4)
For q = l/2,
eqn. (C4) gives the simple result
-nRT nE,,,, - nE O G ___ ~FK”
showing
=
-2RTD,, cc51
Fk”a
that a plot of a!?,,,
versus l/a
will give a straight
line of slope
-2RTD,
d(G,z) d(l/a)
z
(W
Fk”
from which k” is calculable. Equation (0) shows that an extrapolation will give nE O. Experimental verification of this method has appeared elsewhere [lo]. The q = l/4 and q = 3/4 cases of eqn. (C4) lead to n(E,,4-E”)z$
(C7)
1 and
1
n(E,,,-E”)zy respectively. n( E,,, Finally, K O,
n(E,,l
We
On subtraction,
- E,,,)
= F
if we combine arrive
-E”)z
((33)
these last two results
give
m(9) + (31-a iTKl’)rRT
(C9)
eqns. (C5) and (C9) so as to eliminate
the term containing
at
(56.5 mv) - n ( Et,, - E3,4 pa_
y-1
)
(C10)
Thus, our reasoning leads to the prediction of a linear relationship, between nE,,, as Fig. 7 illustrates. The and n( E,,4 - E3,4) in the region of near-reversibility, numerical evidence in Fig. 3 confirms this predicted linearity and shows that it is most extensive for (Y values in the 0.2-0.5 range.
99
The final task in this appendix is to address the question of how large KO must be before the voltammogram is “effectively reversible”. From eqn. (C5) we see that nE, /2 is within 1 mV of its reversible value of nE” provided that -
vRT
< 1.0 mV
28.~”
which translates
(Cl11 to
>40
K”
(C12)
and explains APPENDIX
the first item in eqn. (6).
D
In this appendix we consider simplifying features of steady-state waves at microdisc electrodes when KO is small enough to make the voltammogram either totally irreversible or nearly irreversible. Equations (17) and (18) show that as K o approaches zero, which corresponds to total irreversibility, 8-l
(Dl)
and 2(h + 1)
1-q + T) -+ 4
3h(2h
The latter expression
032) shows that, if H represents
the irreversible
limit of h, then
(D3) Values of H computed from this formula for q = l/4, l/2 and 3/4 are listed in Table 5. As well, this table includes the corresponding values of ln(3rH/2), which are needed later. For future use, we also require values of the quantity defined by (D4)
TABLE
5
Parameters
associated
with the totally
irreversible
hmit
4
H
ln(3vH/2)
l/4 l/2 3/4
0.07252 0.2275 0.7545
- 1.074 0.06973 1.269
M 0.3654 1.125 3.472
To determine these, one starts with an expression and differentiates it. obtaining d(q8) p= dh
from eqn. (14)
6[(h+1)(4h+a)-h(2h+7r)] (D5)
[2(h+l)+3h(2h+7~)]~
Simplifications
M=
for q0 obtained
become
possible
when the limit is taken,
2q(H + 1) (l-q)H[3(1-q)(4H+a)-2q]
leading
to the result (D6)
-I
as the expression for M. Table 5 contains values of this parameter for q = l/4, l/2 and 3/4 calculated with the aid of formula (D6). We now turn to conditions of near-irreversibility and derive an approximate result that permits the determination of (Yand E O. When a voltammogram is close to irreversible, 0 will be slightly greater than unity and h will exceed H by an amount depending on 8. If we regard ln( h) as a function of 8. then by Taylor’s theorem lnh=(lnh)B=,+(8-1)
+ .
However, the smallness the first two, so that In h=ln
of 0 - 1 permits
(D7)
O=I
O=l
us to disregard
right-hand
terms
beyond
H+(B-l)(M+l)
(D8)
where we have incorporated the definitions (17) to replace h in eqn. (DS), we arrive at
of H and
M. If we now employ
(M+I)(@-I)-ln@=ln(K”)-ln(37rH/2)-CXln(8-1)
eqn.
(D9)
after rearrangement. Because 0 is close to unity, In 0 may be replaced by (0 - l), so that the left-hand side of eqn. (D9) becomes M(8 - 1). The expression exp{ nF( E, - E o )/RT } is next substituted for (0 - 1) in both of the terms in which it appears, leading to
=exp{g(EO--E,))[ln(KO)-ln(3aH/2)-g(E,-E”)] after each side of the equation is multiplied by exp( nF( E o - E,)/RT). Equation (DlO) is valid for any value of q. Writing it for the q = l/2 has 1.125 exp
i
gT(
E,,,
@lo) case, one
- Er,)
(Dll)
101
when numerical values are inserted from Table 5. Similarly, one may insert first the q = l/4 conditions into eqn. (DlO), and then separately set q = 3/4. Subtraction of the resulting expressions leads to 0.3654 exp
i
$
( E1,4 - Er))
- 3.472 exp{$$(E3,,4
- Er))
=exp{~(E”--E,)}[2.343--~(L.,;~--~~,~~)] lXF = RT exp {*(E~-E~)}[ RT
60.tmV
-PZ(L.,,~-E~,,~)]
The left-hand side of this equation is identical with the quantity defined eqn. (26). We see that L plotted versus n(E,,, - E,,,) gives an intercept mV)/a and a slope that conforms with eqn. (27) of the main text.
(D12) as L in of (60.2
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