Anal. Chern. 1984, 5 6 , 2909-2914 (4) Blaedel, W. J.; Klssel, T. R.; Boguslaski, R. C. Anal. Chem. 1972, 4 4 , 2030-2037. Brady, J. E.; Carr, P. W. Anal. Chem. 1980, 52, 980-982. . , Hameka, H. F.; Rechnltz, 0. A. Anal. Chem. 1981, 53, 1586-1590. (7) Pedersen, H.; Horvath, C. Appl. Biochem. Bioeng. 1981, 3, 96. (6) Tran-Mlnh, C.; Broun, 0. Anal. Chem. 1975, 4 7 , 1359-1364. (9) Raclne, P.; Mlndt, W. Experientis, Suppl. 1971, 78, 525-529. (10) Meil, L. D.; Maloy, J. T. Anal. Chem. 1975, 4 7 , 299-307. (11) Mell, L. D.; Maioy, J. T. Anal. Chem. 1978, 4 8 , 1597-1601. (12) Durliat, H.; Comtat, M.; Mahenc, J.; Baudras, A. Anal. Chlm. Acta 1978, 85, 31-40. (13) Kernevez, J. P.; Konate, L.; Romette, J. L. Blotechnol. Bioeng. 1983, 25, 845-855. (14) Gough, D. A.; Leypoldt, J. K. Appl. Biochem. Bioeng. 1981, 3, 175-206. (15) Gough, D. A. ; Leypoldt, J. K. Anal. Chem. 1979, 51, 439-444. (16) Gough D. A,; Leypoldt, J. K. J . Electrochem. SOC. 1980, 127, 1278-1 286. (17) Gough, D. A,; Leypoldt, J. K. AIChE J. 1980, 26, 1013-1019.
(18) (19) (20) (21) (22) (23) (24) (25) (26) (27)
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Durliat, H.; Comtat, M. Anal. Chem. 1980, 52, 2109-2112. Feldberg, S. W. Nectroanal. Chem. 1989, 3 , 199-296. Winograd, N. J . Nectroanal. Chem. 1973, 4 3 , 1-8. Heinze, J.; Storzbach, M.; Mortensen, J. J . Nectroanal. Chem. 1984, 765,61-70. Booman, G. L.; Pence, D. T. Anal. Chem. 1965, 37, 1366-1373. Sandifer, J. R.; Buck, R. P. J. Nectroanal. Chem. 1974, 4 9 , 16 1- 170. Hanafey, M. K.; Scott, R. L.; Ridgway, T. H.; Rellley, C. N. Anal. Chem. 1978, 50, 116-137. Lasla, A. J. Electroanal. Chem. 1983, 746, 397-412. Durliat, H.; Comtat, M. Anal. Chem. 1982, 5 4 , 856-861. Richtmeyer, R. D.; Morton, K. W. “Difference Methods for Initial Value Problems”; Interscience: New York, 1967.
R~~~~~~~for review ~~b~~~~~23, 1984, ~ ~ ~ ~ bjUly~ i t t ~ d 24, 1984. Accepted July 24, 1984.
Five-Electrode Thin-Layer Cell for Spectroelectrochemistry Applied to Spectrocoulometric Titrations D. A. Condit,’ M. E. Herrera, M. T. Stankovich? and D. J. Curran* Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003
The spectroelectrochemlcal cell described here Is the first reported whlch is capable of performing dlrect spectrocoulometrlc tltratlons wlthln a thin-layer cavity with negllglble edge effects. Thls was accompllshed by mlnlmlzlng the potential gradlenl and solution resistance effects Inherent to the geometric requirements for slrnultaneous spectral and electrical monltorlng of electrochemically lnltlated processes. Twin reference and counter electrodes and a dual-tapered thlncavlty geometry made a spectrocoulometrlc tltratlon posslMe by decreasing the response tlme whlch, In turn, mlnlmlzed the edge effect. The cell performance characterlstlcs are examined In terms of the devlatlon from theoretlcal thln-layer predictions due to solution resistance and mlnlgrld dimension effects. An emplrlcal cell rate constant of 0.053 s-’ was determined from the spectrocoulometrlc titration of ferrocyanide.
When first introduced (1,2),optically transparent electrodes (OTE) had appeal as a way to dynamically follow heterogeneous and coupled homogeneous reactions a t and near the electrode-solution interface. Titrimetry, with mass transport dependence on convection, came later as an analytical application of OTE’s. Several cells used to perform spectrocoulometric titrations have a standard 1-cm optical path length and a solution volume of a few milliliters. The first cell design reported (3),of this nature, has a tin oxide-coated quartz working electrode which also serves as a cell window while a more recently reported cell ( 4 , 5 )has a quartz or Pyrex cuvette-shaped bottom into which are dipped the counter, reference, and working electrodes. For both cells, the optical path goes through the bulk solution and out of the cells Present address: United Technologies Research Center, Silver Lane, MS-94. E Hartford. CT 06108. Present address: Department of Chemistry, Kolthoff and Smith Halls, 207 Pleasant Street, SE,University of Minnesota, Minneap-
olis, MN 55455.
unobstructed by other electrodes and stirring devices. From titrations performed with these cells, the formal potential, Eo’, the number of electrons transferred, n,and the total amount of material present can be determined. In contrast to these spectrocoulometric cell designs, thinlayer spectroelectrochemical cells utilizing gold minigrid working electrodes have a shorter optical path length, typically less than 0.3 mm, and mass transport is solely by diffusion (6). The large electrode surface area-to-volume ratio presents an opportunity to carry out rapid direct spectrocoulometric titrations within the volume containing the gold minigrid. However, the existence of edge effects a t the minigrid boundaries led to the development of potentiostatic procedures (7,8) to indirectly determine the number of electrons transferred, n, instead of to direct quantitative determinations of the total amount of material electrolyzed. This paper reporb on a five-electrode symmetrical sandwich cell comprised of twin reference and twin counter electrodes, a gold minigrid (333 lines/in.) working electrode, and a small-volume (11kL) thin-layer cavity of special geometry. This cell has a response time sufficiently fast to minimize the edge effect and permit direct spectrocoulometric titrations. Cell performance was evaluated by thin-layer cyclic voltammetry and potential step experiments. Results are examined in terms of theoretical predictions.
EXPERIMENTAL SECTION Instrumentation. Solution resistance measurements were made by using an Industrial Instruments Model RC 16B2 conductivity bridge. The electrochemical instruments used were EG&G Princeton Applied Research (Princeton, NJ) Models 173, 175, and 179 along with a Hewlett-Packard Model 7040A X-Y recorder. A Beckman Model MVI UV/vis spectrophotometerhas used for the spectrocoulometric titration of ferrocyanide. The most sensitive expanded absorbance range is 0.01 A. Computer programs were written in-BASIC and run on an Apple 11+ with 48K memory. Chemicals, Methyl viologen was purchased from British Drug House, Poole, England. All other chemicals used were ACS reagent grade.
0003-2700/84/0356-2909$01.50/00 1984 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
hv
Figure 1. Exploded and assembled (inlet/top view) diagrams of the five-electrodethin-layer spectrocoulometric cell. Note: h v is centered in between reference electrodes. The individual components are given as follows: (A) Ag/AgCI reference electrode (0.25 mm 0.d.); (B) outlet Teflon tubing transfer line (1.7 mm 0.d. and 1.1 mm i.d.); (C) rear C-clamp brass jaw; (D) Vycor frit: (E) reference electrode cell window (6.35 mm thick Pyrex); (F) goM wire (0.25 mm 0.d.) for minigrid contact: (G) Two 50-pm Teflon spacers; (H) gold minigrid (333 lines in.) worklng electrodes; (I) counter electrode cell window; (J) front C-clamp brass jaw; (K) clamping hex bolt (8132);(L) Ag wire counter electrode (0.25 mm 0.d.): (M) inlet Teflon transfer line; (N) brass base (slotted on bottom) of cell holder; (P) brass post of cell assembly holder; (a) standoff machined into front jaw: (R) plastic shim sandwiched between reference window and rear jaw. Spectrocoulometric Cell. A diagram of the cell is shown in Figure 1. The assembly consists of two 20 mm X 45 mm X 6.35 mm optically flat Pyrex windows (Edmund Scientific Co., Barrington, NJ). One window was drilled (3.2-mm diamond hole drill) from the outside to meet a hole (4 mm) drilled 1mm deep from the inside surface. A 1 mm thick Vycor disk was subsequently epoxied in place on each ledge, and the reference cell bodies (3 mm 0.d.) were expoxied in place against the Vycor. The glass reference tubes, bent to about 120°, were directed upward at about a 45O angle to the horizontal plane so bubbles present in the filling solution would rise and not form an obstruction. Female ground glass tops were provided. Inlet and outlet ports were made by epoxying two 1.2 mm 0.d. stainless steel needle tips into 1.4 mm diameter holes so the flat ends were flush with the inside surface and the pointed ends extended beyond the front surface. The latter ends were joined to Teflon tubing (1.7 mm o.d., 1.1 mm id.). The window holding the counter electrode assemblies was prepared by drilling 6.2-mm holes for each electrode. Vycor disks (5.8 mm x 1mm) were epoxied onto the ends of glass tubes which were epoxied into the holes with the frit flush with the inside surface and the bent electrode bodies directed upward at the same angle as the reference electrodes. The reference and counter electrodes were 0.25 mm diameter silver wire soldered to copper posts epoxied into (7/15) female ground glass joints. Silver chloride was coated onto the reference electrode wires with aqua regia. The same filling solution,0.1 M KCl and 0.09 M phosphate buffer (pH 7.00), was used in both the counter and the reference electrodes. Gold minigrid (333 lines/in.) was obtained from Buckbee-Mear, Co., Minneapolis, MN. Strips of Teflon 50 pm thick and 12.5 mm wide were used to make the cavity. Four strips were stacked together and a brass template was used as a cutting guide to cut out the cavity shape. The gold minigrid was placed in the center of the stack. Contact to the minigrid was made with 0.25-mm gold wires as shown in Figure 1. A cell for the reference beam of the spectrophotometer was constructed by using the same plate glass for the windows, but holes were not drilled for electrodes and ports. It was filled by placing a few drops of solution into the cavity containing the minigrid and assembling the sandwich. Cell Holders. The cell holders shown in the inset/top view in Figure 1were machined from brass. The 0.66-cm viewing width and the height of the cell holder post dimensions aligned the gold minigrid correctly in the light beam. The arrangement of clamping bolts, together with plastic PVC standoffs, distributed the
clamping pressure to prevent cracking the windows. The brass post depth between the C-clamp jaws ensured that the tightened assembly was hermetically sealed. The bases of the cell holders were machined to have two channels which fit the optical rails of the spectrophotometer. Spectrophotometric Side Panel. A replacement side panel for the sample cell compartment was made. Stainless steel bulkhead fittings of in. were mounted on it to accommodate the cell inlet and outlet lines. These fittings were electrically insulated from the side panel. A stainless steel Luer-hub syringe needle was coupled to the outlet port. Glass or plastic syringes could then be used to fill and flush the cell. Electrode contact posts on the panel joined leads from the cell to leads from the potentiostat. Determination of Reference Potential and Cell Thickness. A 0.1 M phosphate buffer (pH 7.00) solution, 5 mM in ferrocyanide and ferricyanide, was drawn into the cell from an external flask. The reference electrode potential was determined by using the null potential mode of the PAR 173, and the value vs. SHE was calculated according to the procedure of O’Reilly (9). The result was 290 3 mV at 22 h 2 “C. When the measured absorbance at 420 nm and the known molar extinction coefficient (c = 1020) of ferricyanide were used, a cell thickness of 197 h 2 wm was calculated. Cyclic Voltammetry of Methyl Viologen. Cyclic voltammograms of methyl viologen were obtained to characterize cell performance. The sample was placed in a 10-mL pear-shaped three-necked flask equipped to enable purging with argon to eliminate interference from oxygen reduction (IO). After purged samples were loaded into the cell, the minigrid was preconditioned prior to initiation of the potential sweep by holding the potential at 0 mV for 5 min, followed by a 2-min hold at the -100-mV starting potential. The coulometer was reset and the potential scanned to approximately 100 mV beyond the cathodic peak potential before reversal. Investigations were performed to compare the three- and five-electrode configurations, to determine the effect of scan rate using the five-electrodeconfiguration, and to show the effect of IR compensation at a scan rate of 1 mV/s. Spectrocoulometric Titration of Ferrocyanide. After the cell was cleaned by drawing 2-3 mL of distilled water through it, a similar volume of ferrocyanide was used to fill the cell with sample. The minigrid electrode was preconditioned by holding at 90 mV for 5 min ahd then stepping to 190 mV for 2 min before starting the potential sweep. The wavelength was set to 420 nm. The double-beam autoslit mode of the Beckman MVI was used for monitoring the spectral response. A 0.325-mm slit width and 1-s period were typical. Each spectrocoulometric titration was followed by an identical experiment performed on a blank solution. The absorbance monitored during the blank experiments did not change from the initial base line. RESULTS AND DISCUSSION Cyclic Voltammetry. Theoretically, the cyclic voltammetric peak shape in both sweep directions is Gaussian in a thin-layer cell, and the difference between anodic and cathodic peak potentials is 0 V (11). It is well-known that there are two design problems in real cells which prevent experimental curves from attaining this shape. The first is usually called “resistance effect” and arises from a potential gradient across the face of the electrode which is produced by solution IR drop. The second is called the “edge effect” and refers to mass transport into and out of the thin-layer cavity via the edge connection(s) of the cavity to the rest of the cell. The present cell design attempts to minimize the first effect by using two counter electrodes and two reference electrodes, each pair being connected electrically in parallel. Figure 2 shows a cyclic voltammogram for methyl viologen at a scan rate of 5 mV/s using only one counter and one reference electrode and a second current-voltage curve using the pairs of counter and reference electrodes. The improved performance of the five-electrode configuration is clear, showing a decrease in AE of 75 mV along with improved peak symmetry. The peak distortion apparent in these cyclics is a function of scan rate. Additional cyclic experiments with the five-electrode ar-
ANALYTICAL CHEMISTRY, VOL. 56. NO. 14. DECEMBER 1984 2911 B
A
190
C
390 590
390 590 790
E
VI
190 390 590 790
SHE lmv)
4. Cyclic Mltsmmograms ( a m e s 2) of 4.70 mM (A and B) and 2.35 mM IC) ferrocvanide in 1.0 M LICI. The backoround cvclic voltammograms (cukes I) were -r UM the =;;le wndiions for 1.0 M LICI. I
-IW
I
-300
-500 E v s SHE l m i )
-700
Flgure 2. Cyclic voltammetry f a 1 mM methyl viologen, 0.1 M phosphate buffer (pH 7.00) in 0.1 M KCI. Canparim of a three- (-1 and fhre-(---) electrode configmtion at 5 mVls scan rate.
A
B
0
u
C
Flgun 3. Oold mlnlgr!dlTeffonspacer compo8nBs. Arrangement on the far left is typlcal of a three-eiectr& spectmelectrochemical thirdayer cell. me three adjacent arrangements (A. E, and C) depict the evolution of the fiie-electrode Spectrceoulometrlc cell. The total mlnlgrid edge boundaries for these configurations are (A) 20 mm, (E) 8 mm,and(C)2 mm. rangement a t scan rates of 2 and 1 mV/s produced peak separations of 60 and 40 mV, respectively. Furthermore, when the scan a t 1mV/s was repeated using positive feedback IR compensation, the result was a peak separation of 27 mV. This performance is an improvement over a three-electrode tin oxide OTTLE where cyclic experiments on the same concentration of methyl viologen a t the game scan rates produced peak separations of 275,170, and 100 mV, respectively (12). The cavity arrangement for the above experiments is shown in Figure 3A. Here, a large portion of the cavity volume does not contain the minigrid electrode and two, 10-mm lengths of minigrid edge adjoin space occupied by the solution. As such, the cell is ill-suited for spectrocoulometry because of the large edge effect. In parts B and C of Figure 3, the total minigrid edge exposed to adjoining solutions was sumesaively decreased from 6 to 2 mm, respectively. The electrcde areas (one side) for Figure 3A-C were estimated by measurement to he 0.85, 0.60, and 0.55 cm2, respectively. Figure 4 shows the corresponding cyclic voltammograms for the three cavity arrangements. Curves 1are background scans, and curves 2 are cyclics of ferrocyanide solutions. The cyclic for arrangement A in Figure 3 has the largest peak separation and the current fails to return to the base line, indicating mass transport of ferrocyanide by diffusion from the region of the cavity where the tips of the counter and reference electrodes are located into the region of the cavity occupied hy the minigrid. The cyclic for arrangement B in Figure 3 shows improvements and that for arrangement C shows that the
190
E
390 590 790 VI
SHE ( m v l
Flgure 5. Slmultanews (A) absorbance and (E) current-potential curves for the spectrowubmetrlc tnration of 2.35 mM ferrocyanide in 1.0 M LiCI.
Table I. Resistance of Various Electrode Combinations in 0.1 M KCI. 0.09 M Phosphate Buffer (pH 7.00) resistance. R
electrode combination upper counter/minigrid lower counter/minigrid upper reference/minigrid , Lower reference/minigrid combined caunters/minigrid combined references/minigrid a See Fieure 3.
arrangement Ao
arrange. ment C
4375 4370 4040 4025 2220 2010
12600 11600 9600 6500 6150 4250
current returns to the base line. The concentration of ferrocyanide corresponding to the cyclic voltammograms in Figure 4C is half that of 4A and 4B, yet any contribution to the current caused hy masa transport across the edges of the minigrid is very small and cannot be seen in the cyclic response. The major compromise necessary to achieve this is a modest increase in cell resistance because of the narrow channels. Table I shows the resistances between various combinations of electrodes for the arrangements in parts A and C of Figure 3. The larger resistances of the arrangement in Figure 3C are no problem to the potentiostat in terms of potential control, and IR compensation remains effective. Figure 5 shows the cyclic current-potential and absorhancepotential C U Nrecorded ~ simultaneously for 2.35 mM ferrocyanide in 1 M LiCl a t a scan rate of 1mV/s. There is
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ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
close agreement between the potentials at the inflection points of the absorbance curves and the peak potentials of the current-voltage curves. Data were obtained to calculate the titration efficiency, defined as the ratio of theoretid coulombs to experimental coulombs. The former requires knowledge of the cavity volume, and the latter involves a correction for background coulombs. On the basis of the evidence cited earlier that diffusion to or from the narrow channels (Figure 3C) is negligible, the cavity volume was calculated from the measured cross-sectional area occupied by the minigrid (0.55 cm? and the measured cavity thickness (197 pm) with a correction for the volume occupied by the minigrid of 0.13 p L which is based on the dimensions of the minigrid wires provided by the manufacturer. The result is a cavity volume of 10.7 p L . Experimental coulombs were obtained as the difference in the coulometer readout between test and blank runs. The titration efficiencies for the anodic and cathodic scans were 0.99 and 1.02, respectively. These results demonstrate that coulometric electrolyses can be carried out in this cell. The E" ' of 462 mV was calculated from the average of the anodic and cathodic peak potentials. This E"' is in good agreement with literature values of 461-466 mV (13). The time for a cyclic scan was 20 min. Thus, a direct determination of n or a quantitative analysis, along with a determination of E" ', was carried out in this period of time. Only 10 min is required if the less accurate estimate of E" ' produced by a scan in a single direction is sufficient. These times are considerably shorter than the 70 min required by an earlier POtentiostatic approach to spectrocoulometry (14). Current-Time and Absorbance-Time Response. A rigorous description of controlled potential electrolysis in quiet solution a t a minigrid electrode in a restricted boundary cell is not straightforward for the following reasons: (1)the minigrid electrode is not strictly planar, (2) even if the Cottrell equation applies to the minigrid electrode, it cannot remain valid after the diffusion front reaches the cell wall, (3) at later times in the electrolysis, the current decays exponentially, and (4) the current-time response is dependent on IR effects induced by the electrode thin-layer cavity geometry. Oglesby, Omang, and Reilley (15) considered planar electrodes and used two equations previously derived for the analogous heat-transfer problem (16). In each of these equations, the current-time response involves a sum of exponentials. One of them converges more rapidly at shorter times and the other a t longer times. We have written programs in BASIC to solve these equations for the concentration-distance profiles and to obtain the current-time response. We refer to the program for the short-time equation as the S?TL program and to that for the long-time equation as the L'ITL program. Programs were also written for the Cottrell (C) and exponential decay equations. The parameters used in' the calculations corresponded to the cell and solution conditions used in the experiments: cavity thickness, 100 pm (half of the total cavity thickness with a planar electrode positioned at the midline); diffusion coefficient of ferrocyanide, 6.32 X lo4 cm2/s; initial concentration of ferrocyanide, 2.35 X lob3M; and electrode surface area of 0.42 cm2 (which corresponds to the total surface area of the minigrid wires). The concentration-distance pfofiles based on the Cottrell and STTL equations were indistinguishable for electrolysis times of less than 1 s. After that time, the diffusion front reaches the wall and the plots begin to differ. The L'ITL plot then applies and becomes indistinguishable with a simple exponential (SE) plot for the duration of the titration. Figure 6 shows the calculated concentration profiles for the L?TLand simple exponential (SE) models for electrolysis times from 1to 10 s. Both models yield the same result for electrolysis times greater than about 4 s. A concise summary of the
08
07 06
05 04
03 02 01
'0
20
80
60
40
100
DISTANCE FROM ELECTRODE (,urn)
Figure 6. Comparison of the calculated concentration profiles for the (---) and SE (-) models for electrolysis times from 1 to 10 s.
LTTL
48 46
1 I
\ E ,3 ,2 , @ I
0
1
t
2
1
3
I
4
,
5
,
I
6
7
'
8
I
9
I
,
1
0
TIME OF ELECTROLYSIS ( s e d
Figure 7. Graphical representation of the four thin-layer equations. This plot assigns time regions to the appropriate models. The theoretical curve which predicts the current-time behavior for a restricted boundary thin-layer cell with the same dimensions as the spectrocoulometric cell reported here is 1-3-5.
current-time calculations made from equations given in the Appendix for all the models is shown in Figure 7 where In I is plotted vs. time. Electrolysis would be expected to follow the C o t t r e l l s m equations (region 1)initially. At later times, the L ~ model L holds (region 3) followed by the S E model (region 5). The line labeled 2 is an extension of the SE model to shorter times, and it is seen that the current is underestimated. The extension of the Cottrell equation (curve 4) is clearly inapplicable since it does not take into account depletion of the bulk concentration. The curves in Figure 7 suggest that the Cottrell equation could be used for electrolysis times less than about 3.5 s and the S E equation for times greater than this, thus avoiding the mathematical complexities of the S ~ and L L ~ models. L A simple expression derived in the Appendix predicts the time at which the Cottrell equation fails, tc
tc = L2/4D
(1)
This expression is general since only the thickness of the cavity, L, and the diffusion coefficient are involved. For our cell, tc is 4.0 s. Equation 1is valid only if the minigrid behaves as a plane electrode. The question of the planarity of minigrid electrodes was addressed by Petek, Neal, and Murray who showed that if the ratio A,/A, (where A , is the total surface
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
45
35
1
2913
\
.OO?
0 '
i, , 0
,
\
I
25
I
I
50
75
I
IO0
t (secl
, , , ,
, , ,\, , , , , 10 20 3b 40 50 60 70 80 90 100 t (sec)
Figure 8. In 1 vs. t plot (0)of averaged experimental current-time points describing the cell response to a potential step which oxidized 2.35 mM ferrocyanide in 1.O M LiCl at a diffusion-controlled rate. The theoretical prediction is included (---) to provide a quantitative comparison over the duration of the spectrocoulometric titration.
area of the minigrid wires and A,, is the cross-sectional area of holes and wires) is close to 1,the time to achieve diffusion profile averaging is very short and the Cottrell equation holds (17). They estimated the profile averaging time (PAT) as the time required to produce a profile depth equal to one-half the hole size. For our electrode, A,IA,, is 0.385 and the PAT is 1.5 s. These numbers show that the Cottrell equation is not expected to hold until 1.5 s of electrolysis has occurred and that the experimental current will be less than that predicted by the Cottrell equation for earlier times of electrolysis. The data from four potential step experiments were averaged and are shown as the In I vs. t plot in Figure 8, along with the theoretical plot from Figure 7. Quantitatively, there is little agreement between the plots. Qualitatively the experimental results early in the electrolysis are in agreement with the conclusion reached above that the Cottrell equation does not hold. Another factor affecting the shape of the experimental curve shown in Figure 8 is sohtion resistance effects. This subject has been discussed by Goldberg and Bard for thin-layer cells with planar electrodes (18). Such effects would be expected to extend the time it takes for diffusion profile averaging, delay the diffusion front in reaching the wall, and extend the time required to reach bulk electrolysis conditions. As shown by Figure 8, all of these effects extend the time required for complete electrolysis and contribute to the nonlinear region of the logarithmic plot. The figure shows that the electrolysis is about one-half over before a linear relationship occurs. In summary, the combined effects of the nonplanarity of the minigrid, the restricted cell boundary, and solution iR effects must all be invoked to describe the cell response. Cell Titration Performance. For practical reasons, it is useful to establish a figure of merit to express cell performance. Since the latter part of the electrolysis is expected to follow a simple exponential function, the first-order electrolysis rate constant, p, can be computed for this region. Data from four trials for both the current-time and absorption-time experiments were averaged and used to calculate rate constants from nearly linear segments of these response curves. The results are shown in Figure 9, which is a plot of the rate constant as a function of electrolysis time. The rate constants are sensibly constant beyond about 40 s into the electrolysis. There i s a slight slope which may be a solution resistance effect, but
Figure 9. Plot of first-order rate constants, p I and p A, as a function of electrolysis time calculated from current and absorbance data, respectively, for the spectrocoulometric titration of 2.35 mM ferrocyanide in 1.0 M LiCI.
adherence to an exponential model is satisfactory. There is a di€ference between the values of p computed from the current data and those computed from the absorbance data. The absorbance data are probably more reliable because no correction for background signal was required. The average value of the rate constant between 40 and 96 s of electrolysis for the absorbance-time experiments was 0.045 f 0.02 s-'. When this rate constant is used, the time for 99% electrolysis is predicted to be 111s, which is longer than the experimental electrolysis time of 96 s. However, it is clear from the earlier discussions and from Figure 9 that the electrolysis does not follow a simple exponential model over the entire time of the experiment. Therefore, an average of the rate constants derived from the absorbance data over the entire time of the experiment was considered. This average rate constant of 0.053 s-l predicts an electrolysis time of 94 s which is in good agreement with experiment. T h e rate constant determined in this manner must be regarded as a useful but an empirically derived cell parameter. The achievement of quantitative electrolysis in the spectrocoulometric experiment in less than 2 min represents a significant improvement in cell design.
APPENDIX Thin-Layer Equations. The definitions applied to the thin-layer equations are given as follows: C(x,t) concentration profile = C(X,O)
where x = the distance from the electrode, and t = the time of electrolysis after a potential step is applied. The concentration profile is normalized for the initial bulk concentration at t = 0, so the profiles are expressed as concentration ratios
[
current = I ( t ) = nFAD ac;;,t)]zso where n = the number of electrons transferred, F = Faraday's constant, A = the area of the electrode, and D = the diffusion coefficient of the electroactive species. The current a t ~t= 0 (the electrode surface) is calculated from the concentration gradient (derivative) away from the electrode according to Fick's laws. The four sets of thin-layer equations for concentration profiles and current which apply to different time regions of a coulometric titration are given as follows. C = the Cottrell equation for diffusion-controlled electrolysis
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
2914
where Cobis the initial ( t = 0) bulk concentration. SE = simple exponential equations. The K = 0 term from the LTTL relationships was used to represent the following simple exponential decay equations.
s)
Calculation of the time it takes for the diffusion layer to reach the wall, t ,
I(0,t) = At the wall, when x = L
i.e., the concentration at the wall first becomes less than the initial bulk concentration. To find t ,
2nFADCob ~ X P ( L
(7) (8)
Calculation of the Time after which the Cottrell Equation Is No Longer Valid, t o This calculation assumes that the boundary condition for diffusion-controlled electrolysis must hold; Le., C(L,t)/C(L,O)= 0, where x = L is the electrode surface. This treatment also assumes that the above boundary condition will not hold unless the K > 0 terms are used. Therefore, the time above which the K = 0 term (simple Cottrell case) does not satisfy this boundary condition is tc. When the K = 0 term is used at the boundary condition
By the properties of the error function
L =2 2(Dt,)lI2
L2 t,=---16D
-
When erfc(Y) = 1 - erf(Y) is used and rearrangement occurs
(0.01 cm)2
16(0.632X
cm2/s)
= 0.99 s
short-time thin-layer model. This general equation is for a planar thin-layer cell with the electrode at x' = L and wall a t x ' = 0. Note that x ' is differentiated from x . This is because in C, x = 0 is the electrode surface and in STTL, x ' = 0 is the wall. Thus, x 'is the distance from the wall while x is the distance from the electrode surface. The following concentration ratio and current relationships converge a t times of less than 1 s as applied to the cell reported here. STTL =
C(x - '$1 C(x'0)
1-
K=o
[
+ 1 ) L - X ' + erfc (2K + 1 ) L + x ' 2(Dt)l12 2(Dt)l12
(2K
I(L,t) = nFAC,b( $ ) ' 2 [ i ( - l ) K ex([
1
(3)
g ).
-*
long-time thin-layer model. This general equation is for a planar thin-layer cell with the electrode surface at x = 0 and wall at x = L. It converges at longer times (greater than one second in our studies) LTTL =
-C(x,t) - C(x.0)
When the properties of the error function are used
--L (Dtc)'I2
-2
(0.01 cm)2 t c = -L2 - = 3.96 s 4 0 4(0.632 X cm2/s)
-
C(-l)" erfc
At x' = L (at the electrode)
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RECEIVED for review May 14, 1984. Accepted July 30, 1984. This paper was taken, in part, from the Ph.D. dissertation of D. A. Condit and was presented, in part, at the 34th Pittsburgh Conference and Exposition, Atlantic City, NJ, March 1983.
