Anal. Chem. 1992, 64, 1014-1021
1014
Instrumental, Theoretical, and Experimental Aspects of Determining Thermodynamic and Kinetic Parameters from Steady-State and Non-Steady-State Cyclic Voltammetry at Microelectrodes in High-Resistance Solvents: Application to the fac / mer- [ Cr(CO),( q3-Ph,PC H2CH2P(Ph)CH,CH2PP h2)] + / 0 Square Reaction Scheme in Dichloromethane Alan M. Bond,*st Stephen W. Feldberg,t Howard B. Greenhill, and Peter J. Mahon Department of Chemical and Analytical Sciences, Deakin University, Geelong, Victoria 321 7, Australia Ray Colton’ and Tania Whyte
Inorganic Chemistry Section, School of Chemistry, University of Melbourne, Parkville, Victoria 3052, Australia
Instrumental, experlmental and theoretlcal approaches requlred to quantlfy the thermodynamic and klnetk aspects ol the square reactlon scheme relatlng the hc+’Oand mer+/’ redox couples h the hlgh-mktance solvent dkhkromethane, at mlcrodechockr, urckr both steadystate and fast ucan rate (tramknt) conditions, are presented. fat+, mer+, hc0,and mero represent the faclal and meridional Isomers of Cr(CO),( T$P~~CH,CH,P( Ph)CH,CH,PPh,) In the oxldlzed 17 ekctron (fac+, mer+) and reduced 18 electron (taco,mero) configurations, respectlvdy. A computatbnaHy etflclenl s i m ulatlon method based on the DuFort-Frankel algorithm Is rea* applkd to mkrodectrodes and enables dmulauolrcl to be undertaken for both steadydate and translent vohammetry at electrodes of mkrodkk geometry. The mlnlmal ohmlc drop present under steadydate condltlom enables a lhnlted set of parameters to be calculated for the square scheme. However, data relevant to sped- generated as a product of electron transfer have to be determlned from the translent v o l t a m t r y at fast scans rates. For the latter experhnents, a newly ddgned ekctrochemkal cell was developed along wlth relevant electronic ckcultry to mlnknlze the background cumnrt and “ p w m t e d reddance. The cell contdns two matched worklng microelectrodes (one In the test solutlon and one In the separated electrolyte solution) and a common quarkeference electrode whlch passes through both comparbnenb of the cell. I t b conduded that a Judlclouschoke ol steadydate and translenl technlques, such as those descdbed In thk work, are necessary to characterlze complex reac#on schmos In hlgh-reddance solvents. In tbe example
p”ntedhthI8papwgoodagreementbetweenbothregkns of the mlcroelectrode experlments Is obtalned, although uncompensated resistance stlll appears to Influence the fast scan rate data and thb Is Mkated by an apparent fa8tr rate of electron transfer obtalned vla the use of steady-state voltammetry .
INTRODUCTION The use of microelectrodes has extended the useful time domains under which electrode and chemical kinetics may be ‘Present address: Department of Chemistry, La Trobe Universit , Bundoora, Victoria 3083, Australia. YPermanent address: Brookhaven National Laboratory, U ton NY 11973. On sabbatical leave at Deakin University during &to: beyDecember 1989. 0003-2700/92/0364-1014$03.00/0
investigated by fast scan rate cyclic voltammetry and scan rates in excess of 1 million V s-l are now available. This is a result of the lower R,Cd time constant (where R, is the uncompensated solution resistance and Cd is the double-layer capacitance) and the decreased (ohmic) iR, drop associated with these electrodes (see refs 1-9 for example). Additionally, the range of solvents which may be used for voltammetric studies has been extended.1° However, in highly resistive media such as aromatic and chlorinated hydrocarbons, there are still substantial problems in using very fast scan rate cyclic voltammetry even with microelectrodes. Therefore, steadystate measurements with minimal iR, drop and background current may be inherently more attractive under conditions of high resistance if high precision is required.”-16 In practice, it is intuitively obvious that a sensible approach with microelectrodes would involve a judicious choice of both the steady-state (slow scan rate) and transient (fast scan rate) responses, although because of instrumental and theoretical restrictions this concept has yet to be widely adopted. The dominant mode of diffusion at a microdisk electrode is dependent upon the dimensionleas parameter, p , which is related to the radius of the electrode, r, the scan rate, v, and the diffusion coefficient of the electroactive species, D, as follows:
p = [~F?u/RTD]’/~
(1)
At larger values of p , linear diffusion predominates (p > 171,’’ and it is under these conditions that the maximum amount of qualitative mechanistic information can be obtained for reactions where rate constants are very high. Most fast scan rate studiea reported to date have in fact utilized data obtained exclusively in the experimental regime where radial diffusion may be neglected. However, to obtain experimentally acceptable data in high-resistance solvents which can be used for quantitative calculations, it is necessary to decrease the scan rate and reduce the electrode radius in order to lower the current and therefore minimize ohmic distortion.1° As a result of these constraints, radial diffusion will become important, the theory becomes more complex, and the mechanistic information is reduced because the products of the electrochemical reaction (and the subsequent chemical reaction products) rapidly diffuse away from the electrode.18 The minimum iR, drop actually is achieved at steady state11J2 where the response is time (scan rate) independent, but under these conditions no information on products can be obtained by direct observation of their electrochemistry. Nonetheless, steady-state voltammetry is a powerful method for studying fast electrode processes as the accuracy of the measurements 0 1992 American Chemical Soclety
ANALYTICAL CHEMISTRY, VOL. 64, NO. 9, MAY 1, 1992
is considerably greater than when linear diffusion is present and when the iR, drop is high, as is the case in resistive media. Therefore, under high-resistance conditions it can be argued that it is sensible to obtain certain quantitative data accurately from studies a t steady state, and near steady state, as well as from fast scan rate experiments. This of course implies that a theory needs to be available that incorporates both the linear and radial diffusion components and that suitable instrumentation is available to cope with the problems inherent with both steady-state and fast scan rate measurements. In this paper, we report the characterization of the isomerization in dichloromethane of the fac/mer-[Cr(C0),(v3P2P’)]+/0 system, where P2P’ is Ph2PCH2CH2P(Ph)CH2CH2PPh2,at microelectrodesfrom both the steady-state regime, where radial diffusion is important, with microdisk electrodes of various radii, and at fast scan rates where linear rather than radial diffusion becomes dominant. The fuc/ mer-[Cr(C0)3(v3-P2P’)]+/o system has been recently studiedlg and shown to represent an example of the square reaction scheme.mq21Application of the fast quasi-explicit finite difference (FQEFD) digital simulation procedure,22extended for use on a microdisk electrode geometry, is reported as a means of solving the theory efficiently using a personal computer. Additionally, we report the development of new instrumental approaches for background current and iR, drop correction and emphasize the need to use both steady-state and nonsteady-state measurements rather than solely using either fast scan rate transient measurements or steady-state measurements to determine the kinetics and mechanisms of reactions in high-resistance media.
1
EXPERIMENTAL SECTION Reagents. The Cr(C0),(v3-P2P’)was prepared as described in the literat~re,2*~~ and ferrocene (Fc) (Merck) was purified by sublimation. The dichloromethanewas HPLC grade (Mallinkrodt Australia Pty. Ltd.) and the tetra-n-butylammonium tetrafluoroborate (Bu4NBF4)electrolyte was electrochemical grade (Southwestern Analytical Chemicals Inc.). All solutions were degassed with high-purity nitrogen, and the temperature was maintained at 20 f 1 O C . Electrodes, Cells, and Instrumentation. As well as minimal iR, drop being observed under steady-state conditions, the background current, ib, also approaches zero. Steady-state measurements can therefore be obtained at low concentrations of analyte with only minimal interference from background current, and so access to the fast time regime can be obtained with virtually no interference from either background current or resistance effects. The instrumentation and experimental procedures for steady-state voltammetry are therefore exceedingly simple. In contrast, the background current and iR, drop from both the faradaic and background current intrude on cyclic voltammogramsobtained under low analyte concentrationor high scan rate conditions where linear diffusion is the dominant mode of mass transport of the electroactive species. This background current is usually derived from the charging current, which is related to the double-layer capacitance and, ideally, therefore is so this term is very large directly proportional to the scan if fast scan rate techniques are used. On the other hand the faradaic current, if, is dependent on the square root of the scan rate under linear diffusion conditions.% At higher scan rates the ratio of iflib can therefore become quite unfavorable. Indeed, when ib > if, the background current is the major term causing iR, drop problems and at the same time causes the loss of mechanistic information due to the decrease in resolution of the faradaic current. There are at least three commonly used procedures for minimization of the effects of background current in cyclic voltammetry. (a) The use of high analyte concentrations in excess of the millimolar level has typically been used in fast scan rate experiments. Unfortunately this procedure gives rise to a larger iR, drop problem due to the increased current associated with a higher concentration of electroactive species so that this method is not useful in high-resistance solvents where, if anything, lower
1015
w2
Flgure 1. Cell arrangement for subtraction of the background current, where CE represents the counter electrode and where W1 and W2 correspondto the analyte working electrode and the electrolyte working electrode, respectively.
concentrations of analyte would be preferable to minimize the iR, drop. (b) A second method involves the post experimental subtraction of a blank electrolyte solution obtained under identical conditions to the analyte/electrolyte voltamm~gram.~~ Usually these voltammograms are recorded digitally with a fixed current “window”,and under this subtraction procedure the background current proportion is a wasted component of this window, although this problem can be minimized using an analogue ~ f f s e t .A~ further disadvantage of this method is that this subtraction procedure doubles the recorded noise. (c) The third common procedure for background correction is the use of dual cells and the measurement of the difference in current between the analyte/electrolyte cell and the electrolyte ell.^**^^ For this to be successful there are three important conditions that must be satisfied:28 (1)the concentration of supporting electrolyte must be identical in each cell, (2) the electroactive analyte must not significantly influence the double-layer capacity, (3) the working electrodes must have identical area and surface characteristics. Matching of the solid platinum microelectrodes is a difficult but not insurmountable task. Fortunately, the use of microelectrodes substantially reduces the problem of achieving equal potential control at both electrodesbecause of the decreased ohmic iR, drop, and a new method based on this principle was developed for the present study. The new electrode configuration developed for the subtraction of the background current in the fast scan voltammetry experiments in this work is shoyn in Figure 1. The potential of the platinum pseudo-reference electrode was frequently calibrated against the reversible potential for the oxidation of ferrocene.,O The newly designed cell includes a compartment separated from the test solution, but containing solvent and supporting electrolyte alone. The common platinum wire connecting both compartments of the cell ensures that the potential at both platinum working microelectrodes is equivalent. The working electrodes were a matched pair of 25-pm-radius platinum electrodes, which were constructed from 25pm-radius platinum wire (Goodfellows Metal) and sealed in soda glass. These electrodes were polished to expose a cross-sectionaldisk geometry using a range of emery papers and alumina, the finest being 0.05-pm alumina. A group of 0.3-, 0 5 , 1.0-, 2.5, and 5.0-pm-radius platinum electrodes was also constructed by this method for use under steady-state ~0nditions.l~ Steady-state measurements were also conducted in a two-electrode configuration using a Ag/AgCl (saturated LiC1) in dichloromethane reference electrode. The triangular voltage wave form used in cyclic voltammetry was applied between the working Pt microdisk electrode and the reference electrode using a PAR Model 175 Universal Programmer. The current produced under steady-state conditions was measured using a battery-powered Keithley Model 614 electrometer. Under fast scan rate conditions the current difference between the two working platinum disk microelectrodes was measured using the circuit shown in Figure 2 in combination with the cell described in Figure 1and discussed above. The circuit was constructed from a high-performance monolithic quad op-
1018
ANALYTICAL CHEMISTRY, VOL. 64, NO. 9, MAY 1, 1992
I1
V=(ll-I2)R
1 1
12
I Rl
Flguro 2. Circuit for the subtraction of the background current. R , = io5, ioe, or io7 R.
-5 -10
I
I
t
-20
1
-25
1 I
-35' 1000
\ I '
'
'
'
'
""I
'
'
"
10000
"' 100000
'
'
'
""'
1000000
rRLQLLYCr ( I k
Figure 9. Bode plot from the current difference circuit of Figure 2 with a galn of 10' V A-' (Le. R , = I O e Q).
erational amplifier OPA404 (Burr-Brown) and was battery powered to reduce noise. The circuit contains two current to voltage converterswith variable gain (loS-107 V A-l) produced by changing the feedback resistors, Rf(l@-107Q). The resistors were matched in order to produce identical voltages at the outputs of the initial stage; the difference between the two voltages was then obtained at the output of the final stage. The final stage was operated at unity gain in order to maintain the widest possible bandwidth. The output from the current transducer was recorded on a Gould Digital Storage OscilloscopeType 4035 interfaced via an WE488 GPIB to an Olivetti M24 personal computer. The current difference measuring device has a second-order component as can be seen in the Bode plot in Figure 3. Problems with the second-order component may be avoided by using scan rates much lower than the frequency where any significant nonlinearity would be introduced. This maximum scan rate available for use could be estimated from the followingequation:31 Y 5 f/5n (2) where f is the cut-off frequency and n is the number of electrons in the reaction. An alternative approach used was to correct for the nonlinearity by converting the voltammogram into the frequency domain via a fast Fourier transform and applying a correction based on the frequency response of the amplifier?2 An inverse transform is then used, and the corrected voltammogram is generated. Due to the dependence of the current on the scan a reduction in the gain must be used at fast scan rates in order to avoid saturation of the amplifier. This has the benefit of increasing the frequency response of the amplifier, thereby enabling higher scan rates to be used. Consequently, the first method can be applied in most instances under the conditions used in this work. A faraday cage was employed for all experiments, and an advantage of the differential input of the circuit in Figure 2 was the reduction of environmentally induced noise. The uncompensated cell resistance in dichloromethane(0.5 M BhNBFJ was measured using an ac impedance bridge (Wayne Kerr) at a frequency of 10 kHz on a solution of supporting ele~trolyte.3~ The measured resistance (22 000 ohm) compared favorably with the calculated
value of 23 100 ohm obtained from the table of specific resistivities from ref 34 and applying the following equation:36 R, = p/4r (3) where p is the specific resistivity. The anion in our experiments was BF,-, whereas in ref 34 it was C104-. However, from the results, the nature of the anion does not appear to be significant with respect to the uncompensated resistance. The double-layer capacitance was also measured using the impedance bridge, and found to be 180 pF, which is comparable with the value of 185 pF determined from the capacitive current, i, from eq 4 at the 25-pm-radius platinum working electrode. i, = Ycd (4) Extraction of the Faradaic Current Component from the Total Current and Correction for the iR, Drop. The use of the current subtraction circuit developed in this work enables improved resolution of the faradaic signal from the total current, but in the presence of significant iR, drop the result will not be the actual faradaic current. The voltammograms at fast scan rates will have been affected by both the uncompensated solution resistance and the double-layer capacitance to a varying degree dependent on the magnitude of the total current and the cell time constant (R,Cd).*= If one considers that the background current may not be totally due to the current that flows as a result of charging the double layer (i.e. as in the c a e of impurities or due to the solvent limit) then the faradaic current at each potential can be calculated after taking into account the various contributions to the current. The voltammogram of the solvent containing electrolyte known commonly as the "blank" will have current contributions from the capacitive current and possibly faradaic currents from other electroactivespecies present either in the solvent or the electrolyte species present either in the solvent or the electrolyte. For the following discussion i, will symbolize the residual current which is equal to the blank current minus the capacitive current component, and this will contribute to both the blank and the analyte voltammograms equally. Therefore the current observed for the blank will be as follows: i b l d = i, + i, = i, - Cd dE'/dt (5) where dE'Jdt is the scan rate distorted by the presence of R, and Cd and is obtained after consideration of the following equation: E'=E-vt+RuibM (64 where E is the applied potential and E'is the actual potential as a result of ohmic losses. From eq 6a, the distorted scan rate is dE'/dt = -Y + R, dibhk/dt and when combined with eq 5, the capacitance at each potential can be calculated as follows: (7) If Cd is potential dependent, then the actual cd(E) can only be measured if i, is known and this will seldom be the case. The value of Cd can be assumed to be constant in most cases;37this is because Cd is unlikely to vary greater than the uncertainty in the determination of Cd over the potential range required for the characterisation of an electroactivecouple in nonaqueous solvents. Also, it is inadvisable to subtract blank voltammograms where the background current varies substantially with potential due to the large errors introduced and therefore in the majority of cases diblanl/dtwill be minimal, this means that eq 7 will reduce to eq 4 without any need to consider the errors introduced by R, and Cd. In the analyte, there will be the following contributions to the total current imd = if + i, + i, = if + i, + Cd dE"/dt (8) where E"is the actual potential at the electrode and is affected by R, as follows: and the resultant scan rate is
0.08
0.08 (b)
* 2
-
0.00
0.00
1
0
t-.
6
c1
a -0.08
-0.16
0.20
0.00
-0.20
-0.40
-0.60
-0.16 0.20
0.00
-0.20
-0.40
-0.60
-0.40
-0.60
POTENTIAL VI Fc'lFc (VI
POTENTiAL Vs Fc+iFc (VI
0.08 IC)
-.
,-
0.20
0.00
-0.20
-0.40
-0.60
-0.80 0.20
0.00
-0.20 POTENTIAL Vs Fc'lFc
POTENTIAL VI f'c'IFr (VI
(VI
Figure 4. Voltammograms obtained for the oxklatlon of 2 mM Cr(C0)3(+P2Pr)In dlchloromethane (0.5 M Bu,NBF,) with a scan rate of 2000 V s-'. (a) Total current that results from adding b and c. (b) Current obtalned from the supportlng electrolyte. (c) Current difference measured via use of the cur~entdmerence ckcult described in Figwe 2. (d) Orlglnal data wkh cuve (---) belng data as in c and curve (-) being data corrected for IR, distortion uslng the method described in the text.
When combined with eq 8, an expression for the faradaic current is obtained if = ihM - i, - v c d + R,Cd di,,/dt (11) and VCd can be converted to i, based on the previous assumptions regarding the distortion (or lack of distortion) on the scan rate in the blank. It then follows that if = ihd - i, - i, + R,Cd di,,/dt (12) if = ia,
+ R,Cd
di,,/dt
(13)
where idteis the total current minus the current due to the blank as given by eq 5 and is directly measured by the current difference circuit. The derivative of the current-time curve (di,,/dt) was achieved through the use of a Savitsky-Golay quadratic leastsquares operation.39 The minimum number of points used in the calculationwas seven. The number of points was varied so that the time interval of the convolute (Le. the product of the number of points and the time increment between points) was greater than the R,Cd time constant of the cell. If this condition was not satisfied then wild oscillations appeared in the corrected data. The potential axis was then corrected for ohmic drop according to eq 14. A related procedurea requires that the scan rate at each potential be calculated after adjusting the potential scale for the iR, error as for eq 14 and adding a current component at each potential equal to the product of the double-layer capacitance and the difference between the applied scan rate and the actual scan rate (Le. cd[v - dE"/dt]). This procedure was also investigated but was found to be more difficult to implement in the presence of noise. The need for the background current subtraction procedure and iR, correction at fast scan rates even when using microelectrodes is presented in Figure 4a-d. At a scan rate of 2000 V s-l there is substantialbackground current which causes sign5cant
distortion of the voltammogram (Figure4a). Additionally, under these non-steady-state conditions there is considerable ohmic iR, drop distortion. The voltammogram from the cell containing only electrolyte is shown in Figure 4b, and it can be seen that there exists slight nonlinearities in the shape of this voltammogram which makes it necessary to ensure that the background current is properly subtracted. The differential current is plotted in Figure 4c using the same scale as for Figures 4a and 4b to show what proportion of the current was due to the electrochemically active species. In Figure 4d the corrected current is plotted in comparison with the differential current and shows the degree of ohmic distortion that is introduced under the conditions that the voltammogram was recorded. The correction procedure increases the noise because of the step involving the differentiation of the total current-time curve (eq 13) and the slight ripple in the corrected voltammogram is a result of this. However, the voltammogram finally obtained by these procedures is of a quality required for quantitative studies employing theories which assume the absence of background current and iR, drop. At scan rates greater than 2000 V s-' the quality of the data deterioratesrapidly with scan rate since the background current becomes significantly larger than the faradaic current (millimolar concentrations of electroactive species), and the correction for iR, drop becomes extremely large. In this paper we have therefore confined ouraelvea to the report of data obtained at scan rates 52000 V s-l.
THEORY Simulations at a Microdisk. A previously described FQEFD simulation method for cyclic voltammetrynA1 utilizes the DuFort-Frankel algorithm.42 The DuFort-Frankel algorithm is eady adapted to the two dimensional grid required to simulate diffusion-kinetic phenomena at a disk electrode. We used a nonlinear grid which effected a more detailed treatment of the disk edges-the method is conceptually similar to the conformal mapping approach described by Michael, Wight", and Amatorea who used the Hopscotch algorithm which is similar to the DuFort-Frankel algorithm.
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 9, MAY 1, 1992
Table I. Theoretical Results from the Steady-State Simulation (Scan Rate 10 mV disk radius (pm)
k, =
0.3
E112
0.5
- E1/4 E112 E3/4 - El/4
1.0
E112
E314
E3/4
2.5
- E1/4
E112 E3/4
Ell2
5.0
E3/4
- El/4
m
(cm 8-l)
8-l) of
the Square Scheme"
k, = 10 (cm 8-l)
k, = 1 (cm 8-l)
-0.087
-0.080 0.066
-0.088 0.056 -0.090 0.056 -0,093 0.056 -0.103
0.057 -0.090
0.057 -0.094
0.057 -0.103 0.056 -0.111
0.055 -0.110 0.054
0.054
-0.085 0.063 -0.091 0.060 -0.101
k, = 0.1 (cm 8-l) k, = 0.01 (cm s-l) -0,029 0.105 -0.049 0.097 -0,069 0.084 -0.090
0.057
0.070
-0.109 0.055
-0.103 0.062
0.081
0.118 0.056 0.117 0.021 0.116 -0.023 0.111
-0.056 0.101
oParametersEom = -0.085 V,Eom = -0.310 V,and EK = -0.122 V vs Fc+/Fc),other parameters as given in Table I11 with k, = klm = k,m and L Y ~ LYE 0.5. The FQEFD method can efficiently handle the wide dynamic range of homogeneous rate constants required in a complete characterization of a complex mechanism such as the square scheme. The simulation was programmed in QUICK BASIC Version 4.0 (Microsoft)and CPU times (on a 8086 equipped with a 8087 math co-processor) range from seconds to hours per simulation depending upon the input parameters. In contrast, simulations using explicit finite difference can be unacceptably lengthy (see ref 22 for a comparison of computational efficiencies).
RESULTS AND DISCUSSION Cr(C0),(q3-P2P')is soluble and stable in dichloromethane. In this solvent and at 20 OC the complex exists as an equilibrium mixture of the facial (fac-Cr(C0),(q3-Pp')) and the meridional (mer-Cr(CO),(q3-PzP'))isomers (abbreviated to faco and me+, respectively) with the facial isomer being thermodynamically favored by a considerable amount.lg Both neutral isomers can be oxidized to form the corresponding cations, fac' and mer+, but the equilibrium concentrations of these two isomers are now ~imi1ar.l~Thus the electrochemical scheme written in the reduction sense can be depicted as follows.
fat+ + e-
-
mer+ + e-
faco
me@
The square scheme may be represented via the scheme in eq 16.
A+e-
eB
=
(16)
D+eC When all rate constants are infinitely fast on the time scale of the voltammetric experiment, then the system is in apparent equilibrium with an equilibrium potential, EK, dependent upon the equilibrium constants" a~ follows:
As the scan rate increases (transient voltammetry) or the electrode size becomes sufficiently small (steady-state vol-
tammetry), the isomerization kinetics may be "outrun" and the reversible potential of the system will shiftto the potential of the AB couple i.e. Eom Between these limits and under fast scan rate conditions where linear diffusion terms are present, so that information on the products of the reaction become available, it may be possible to identify all four species if the equilibrium constants are favorable, and at this stage it may be possible to estimate EoDC." In order to completely characterize the Cr(C0),(q3-P,P') system it is necessary to determine the standard potentials and the heterogeneous charge transfer rate constants of both the fac+/facoand the mer+/merOredox couples as well as the chemical kinetics and thermodynamics of the isomerization steps fa& + mer+ and faco + mer". The cross reaction fac+ merO +faco mer+ is not included in the theory since it is unlikely that it is important when electron transfers are reversible and the initial condition is at equilibrium.41p* Under conventional conditions with an electrode of radius 0.5 mm and a scan rate of 200 mV s-l only the equilibrium process is observed and therefore only E K is directly measurable. The value obtained is -0.122 V vs Fc+/Fc.lg Experiments at the Steady State. To determine the kinetics of all steps and to calculate the thermodynamics of the proceeses, it becomes necessary to perturb the equilibrium observed with conventionally sized electrodes and low scan rates. With the use of microelectrodes, as noted in the introduction, the simplest way to minimize problems associated with background current and iR, drop, and to enter the kinetic regime, is to use the radial diffusive properties of these electrodes found at the steady-state or, more correctly, near-steady-state regime observed at low scan rates. Under theae conditions, the influence of the reaction products of the fac+ + mer+ reaction following electron transfer is a function of the parameter k?/D (where k is a fiiborder homogeneous chemical rate constant): when this parameter is small (small k and/or small r ) the reaction layer will be larger than the diffusion layer and the kinetics will have a diminished effect; when the parameter is large (large k and/or large r ) the reaction layer will be smaller than the diffusion layer and the kinetics will play a significant role. Figure 5 shows experimental data obtained at electrodes of variable radii under steady-state conditions for the Cr(C0),(q3-P,P') oxidation process at a concentration of 5 X lo-' M. Both the background current and iR, drop are relatively unimportant in these experiments, and this is in contrast to the case with transient experiments. It c h be seen that the isomerization step becomes less important, the smaller the electrode radius becomes, as concluded by noting that a decrease in the electrode radius causes a shift in Ell, toward Eofae+lfae~ under steadystate conditions. This expected shift is demonstrated theoretically in Figure 6 and Table I, from the results of digital simulations using parameters which include the experimental results reported previously from fast scan rate data obtained
+
+
ANALYTICAL CHEMISTRY, VOL. 64, NO. 9, MAY 1, 1992 0.10
I
I
-0.10
1019
Table 11. Experimental Steady-State Results for the Oxidation of Cr(CO)a(qa-PzP’)in Dichloromethane (0.6 M Bu,NBF,) at 20 O C as a Function of Electrode Radius at a
Platinum Disk Microelectrode’ -030
disk radius (pm) -0.50 -0.70
-0.90
. ‘
-1.10 0.15
I 0.05
-0.05
-0.15
-0.25
-035
POTENTIAL Vs Fc‘iFc (V)
Ftgwr 5. Experimental steady-state voltammograms obtained for the oxidation of 0.5 mM Cr(CO)3(q3-PPP‘)In dichloromethane (0.5 M Bu4NBF4). Electrode radli: (a)0.3, (b) 0.5, (c) 1.0, (d) 2.5, and (e) 5.0 pm. 0.10 -0.10
-030 5
.
-0.50
f
-0.70
,
1
t
I1 I
L
0 15
0.05
-0.05
-0 15
-025
-0.35
lPOllhll\l L, l i * l r c ( V )
Flgurr 6. Steady-state voltammograms (scan rate 10 mV s-’)simulated using the parameters summarlsed in Table I with k , = cm s-’. Electrode radll (left to right) 0.3, 0.5, 1.O, 2.5, and 5.0 pm.
under predominantly linear diffusion conditions.lg Steady-state voltammograms are also very sensitive to electron transfer kinetics when the electrode is very small and the advent of nonreversible electron transfer kinetics may cause a shift in El12to a value well removed from the actual Eo value of the fac+/fucocouple as can be seen in Table I. Measurement of the Tomes criterion of E314 - El14indicates the onset of quasi-reversibleor irreversible electron transfer and is useful in estimating the shift in potential due to the electrode kinetics. The position of ElI2is dependent on the sum of the shift due to the contribution of the square scheme and also the shift due to the electrode kinetics. It is possible to estimate the reversible half-wave potential, Erllz,in the presence of quasi-reversible electrode behavior, but in the absence of chemical reactions coupled to electron transfer from the working curves from Figures 2 and 3 of the paper by Oldham et al.,4eusing the difference of E314 - ElI4and knowing the charge transfer coefficient,a, the estimation of Ell2- Ell2 is readily found. The data obtain in this work under steady-state conditions lead to the conclusion that the electrode kinetics are fast. The problem with using slow scan rates to obtain steadystate conditions at a solid microelectrode is that the electrode surface state may vary during the course of the experiment, leading to a dependence on electrode pretreatment and lower reproducibility. Experimental data obtained over the concentration range of 5 X lo-’ to 2 X M as a function of radius are given in Table 11,and the errora reflect this problem. Thus, a decrease in the expected reproducibility of S f 1 mV for data obtained when iR, drop and background current are negligible is found. The dependence on the electrode pretreatment and conditions mean that typical errors of f3 mV
vs Fc+/Fc)
0.3 0.5 1.0 2.5 5.0
-0.083 -0.080 -0.093 -0.106 -0.117
50.0
-0.1226
E3/4
-E l14
(v)
0.073 0.066 0.070 0.068 0.066
’Scan rate used with all electrodes was 10 mV s-l except for the 50-pm electrode where it was 100 mV s-l. Concentration of Cr(CO)3(q3-P2P’) was varied over the range 5 X lo-” to 2 X M. “his potential was calculated from the average of the peak potentials of the voltammogram obtained under conditions where linear diffusion is dominant. in Ellz and f 5 mV in E314- EIl4were found for the steadystate measurements. Despite some limitations with respect to reproducibility, experimentally the data obtained with very small electrodes of radius
