Anal. Chem. 1993, 65,1888-1892
1888
Theory and Application of Potential-Step Transmission Chronoabsorptometry of Long-Pathlength Spectroelectrochemical Cells: Single Reversible Electrode Reaction Qingji Xie, Wanzhi Wei, Lihua Nie, and Shouzhuo Yao' Department of Chemistry and Chemical Engineering, Hunan University, Changsha 410082, Peoples Republic of China
Theory of chronoabsorptometry of a single reversible electrode reaction after a potential-step perturbation using a longpathlength spectroelectrochemical cell is presented. Assuming that the oxidized species is the only absorber at the monitored wavelength, three cases, thin-layer diffusion for Red * Ox + ne and semiinfinite diffusion for Ox + ne Red and Red + Ox + ne, are examined by computation of the analytical solutions or digital simulation. All the theoretical results are verified by experiments.
INTRODUCTION Spectroelectrochemistry (SEC) has been proved to be an effective approach to the study of the redox chemistry of inorganic, organic, and biological substances.lY2 Theory and application of spectroelectrochemical cells with optically transparent electrodes (OTE) in which the optical beam is perpendicular to the electrode surface have been investigated in many papers, and a simple equation linearly connecting the absorbance with the concentration distribution is given as follows1
A = cJouC(x,t) dx
(1)
where w is the width of the solution layer through which the light passes, t is the molar absorptivity, and C(x,t) is the concentration a t a distance x from the working electrode surface at time t. However, low optical sensitivity limits its wider use because of the short effective optical pathlength which is only equivalent to the diffusion layer. In 1979, Tyson and West3p4proposed a new spectroelectrochemical cell in which a narrow light beam passed at grazing incidence over a planar working electrode surface and, therefore, the sensitivity is enhanced due to the long optical pathlength. For a simple diffusion-controlled oxidation reaction with the generation of an absorbing product, a theoretical formula deduced from averaging absorbance was proposed A , = 2t&W-'(D"f)1'2C*~
(2)
where 1 is the optical pathlength over the electrode surface, * To whom correspondence should be addressed.
(1) Kuwana, T.; Winograd, N. In Electroanalytical Chemistry; Bard, A. J.; Ed.; Marcel Dekker: New York, 1974; Vol. 7. (2) Heineman, W. R.; Hawkridge, F. M.; Blount, H. N. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1984; Vol. 13. (3) Tyson, J. F.; West, T. S. Talanta 1979, 26, 117. (4) Tyson, J. F.; West, T. S. Talanta 1980, 27, 335.
w is the width of the investigated solution through which the incident light beam passes, to is the molar absorptivity of the oxidized species, DRis the diffusion coefficient of the reduced species, t is the time, A , is the absorbance of the oxidized species, and C*R is the bulk concentration of the reduced species. Since, then many similar designs of long-pathlength cells have been proposed and applied to studies in spectroelectrochemistry."g Although the possible error of averaging absorbance rather than transmittance was considered, it was concluded that the error was rninimal.6~~ In fact, because of an inhomogeneous distribution of absorbers in the solution adjacent to the electrode surface after an electrochemical perturbation, a quite different analytical spectroelectrochemical model for long-pathlength cells, i.e., that of averaging transmittance rather than absorbance, was discussed compared with the normal OTE cells.1b13 These papers have emphasized the importance of averaging transmittance for long path length spectroelectrochemistry for the case of flow-by and time-dependent systems. In this paper, based on the precise spectroelectrochemical model of long path length cells, the theoretical behavior of chronoabsorptometry for a single reversible electrode reaction after a potential-step perturbation is investigated by digital simulation and computation according to the analytical solution. Three cases are dealt with, and the theoretical results were verified with experimental results for the Fe(CN)e3-/Fe(CN)& system.
THEORY The two spectroelectrochemicalarrangements, i.e., the OTE case and the long path length case, provide two different spectroelectroanalytical models. For a situation where absorbance is measured perpendicular to the working electrode surface, Le., the OTE case, because the light front always reaches a hypothetical solution layer with homogeneity, eq 1 is easily obtained by dividing the investigated solution layer into increments with uniform intervals and using the Lambert-Beer law in each. However, for the situation where absorbance is measured parallel to the working electrode surface, both the solution layer and the incident light beam have to be divided into local increments in order to utilize the Lambert-Beer law for each small incident light beam to correspond to a small (5) Pruiksma, R.; McCreery, R. L. Anal. Chem. 1979, 51, 2253. (6) Rossi, P.; McCreery, R. L. J . Electroanal. Chem. 1983, 151, 47. (7) Zak, J.; Porter, M. D.; Kuwana, T. Anal. Chem. 1983, 55, 2219. (8) Jan, C. C.; Lavine, B. K.; McCreery, R. L. Anal. Chem. 1985,57, 752. (9) Porter, M. D.; Kuwana, T. Anal. Chem. 1984,56, 529. (10)Brewster, J. D.; Anderson, J. L. Appl. Spectrosc. 1989, 43, 710. (11)Fosdick, L. E.; Anderson, J. L. Anal. Chem. 1988, 60, 156. (12) Fosdick, L. E.; Anderson, J. L. Anal. Chem. 1988, 60, 163. (13) Wei, W.; Xie, Q.;Yao, S. J. Electroanal. Chem. 1992, 328, 9.
0003-2700/93/0365-1888$04.00/0 0 1993 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 65, NO. 14, JULY 15, 1993
solution layer, which can be assumed to be of uniform concentration. If the incidence intensity is uniform everywhere, then the absorbance related to the concentration distribution is10J3
A = log w - log ~owlO"C'z~L' dx ... CLO"~~
log m - log
of the concentration distribution of the product, here Ox, has been found, although that of the reactant, here Red, for a similar case has been given which is in a complicated form.16 However, the digital simulation method reported by Feldberg" is quite useful for obtaining the concentration distribution. The concentration distribution can be expressed by Co(l, k + 1)= Co(l, k) + 2DRC,(1, k) +
m
E
(3)
D0[c0(2,k) - C0U, k)l
r=l
where m is the total number of volume elements into which the solution layer, through which the light passes, can be divided, Ci is the concentration in the ith volume element, and i varies from 1to m. It is obvious that the equation connecting the absorbance with the concentration distribution for the long path length spectroelectrochemical cell is quite different from that for a normal OTE cell. Only when the absorbance is small enough can the approximation formula, 10-A = 1-A In 10, be adopted, with a relative error smaller than about 3% for an absorbance smaller than 0.1. Equation 3 can be expressed as the approximation equation which was reported in refs 3,4, and 14.
A = dw-' c C ( x , t ) dx
(4)
However, if eq 4 is used for the interpretation of the spectroelectrochemical information, the most obvious advantage of the long path length spectroelectrochemical cell, i.e., high sensitivity, will be diminished since the observed absorbance must be limited to a small value. Moreover, low absorbance gives poorer precision and accuracy because the relative determination deviation in the absorbance range smaller than 0.2 is far larger than that in the 0.2-0.8 au for spectroscopic investigation using a conventional spectrophotometer.16 In this paper, potential-step transmission chronoabsorptometry of a long path length spectroelectrochemical cell is presented under thin-layer diffusion and semiinfinite diffusion after a reversible electrode reaction takes place. For convenience, it is assumed that only the oxidized species Ox absorbs a t the observation wavelength. Three cases are discussed as follows. Case 1: Ox ne + Red; Semiinfinite Diffusion. For this case, by solving Fick's differential diffusion equation considering corresponding initial and boundary conditions by means of Laplace transformation, the concentration distribution of the oxidized species is given by16
+
~ , ( x , t )= c,* e r f [ x / ( 2 m ) 1
(5)
where Co* a d Do are the bulk concentration and diffusion coefficient of the oxidized species, respectively. By substituting eq 5 into eq 3, A vs t behavior can be obtained. Ox + ne; Semiinfinite Diffusion. By Case 2: Red using a method similar to that used for case 1,the concentration distribution of the oxidized species is Co(x,t) = (DdDo)1/2CR* erfc[x/(2m)l
(6)
By substituting eq 6 into eq 3, A vs t behavior can be obtained. Case 3: Red * Ox + ne; Thin-Layer Diffusion. Since the thin-layer boundary condition expressed as eq 7 aC,(w,t)/ax = -aCR(w,t)/& = 0
t
>0
x =w
1880
(7)
is available for this case it seems that no analytical solution (14) Xie, Y.; Dong, S. J. Electroanal. Chem. 1990,284, 279. (16)Marczenko, Z.Spectrophotometric Determination of Elements; Dizhi Publishing House: Beijing, 1983; Chapter 1 (in Chinese). (16) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: f i n d a mentals and Applications; Wiley: New York, 1980.
Coo', k + 1)= Coo',k) + Do[Co(j + 1,k )
(8)
+
COO'- 1, k) - 2C0(.i, k)]
1 < j < m (9)
Co(m,k + 1)= Co(m,k) + bo[Co(m- 1,k ) - Co(m,k ) ] (10) where 1,j, and m in brackets represent the first, jth, and mth
volume element, k and k + 1 represent the time kAt and (k l)At, for example, Coo',k 1)represents the concentration of the oxidized species in the j t h volume element at the time ( k + 1)At. bo= DoAt/(A.x)2and f ) =~ DRA~/(PX)~ are the dimensionless diffusion coefficients of the oxidized and the reduced species, respectively, and the larger one of them is set to be 0.45 in our programs. Thus, the concentration distribution can be obtained and then the theoretical spectroelectrochemical behavior can be derived from eq 3. Moreover, digital simulation is also used in case 1 and case 2 for comparison with the theoretical spectroelectrochemical behavior obtained from the analytical solutions.
+
+
EXPERIMENTAL SECTION Fabrication of the Long-Pathlength Spectroelectrochemical Cells. The spectroelectrochemicalcell for semiinfinite diffusion was fabricated as described6 with some improvements. The thin-layer spectroelectrochemical cell, shown in Figure 1, was modified according to ref 7. It is very important for the experimental verification of the spectroelectrochemical model mentioned above to obtain a strict rectangular solution through which the light passes, thus procedures for the fabrication are as follows. A plate of glassy carbon, which served as the working electrode (WE) in experimenta, was fixed in a Teflon plate using epoxy resin, and the faces were then polished with Ol#, 03#,04#, OS#,and 06#sandpaper (Shanghai SandwheelFactory) and then with 0.3 and 0.5 pm alumina polishing adhesives successively until plane and smooth surfaces were eventually obtained. Another Teflon plate, which was equipped with the auxiliary electrode (AE)using epoxy resin for a cellfor semiinfiite diffusion and just a single plate for a thin-layer cell, was also polished in a similar way until ita surfaces were very plane and of the same dimensions as the WE plate. The surface dimensions of the WE were ca. 1 X 0.7 cm while those of the WE plate were ca. 1 X 3 cm. All the surfaces, especially the electrode surfaces, were then washed with 0.5 M HzSO,, water, ethanol, acetone, and doublydistilled water successively. For the thin-layer cell shown in Figure 1, the WE plate was linked to the opposite Teflon plate by three screws, and with two thin polyethylene spacers between the two plates, a thin-layer space was formed. The screws were then adjusted carefully to ensure that the width of the lighttransparent region was uniform, which was monitored by an opticalmicroscope with a determination precision of 2 pm (Beijing Optical Instrument Factory). The width of the cell was measured simultaneously. Two pieces of opaque tape were stuck on the front piece of glass to form the front light window with a height equal to ca. 0.2cm, and another two were stuck on the back piece of glass to form the back light window with the same height. The two pieces of glass were then stuck on the side surfaces of the two plates with 704 silicone rubber adhesive (Jiangshuwuxi Adhesive Plant). The distance between the front light window and the back light window was 1.02 cm in this work. In order to minimize nonuniform current density on the working electrode (17) Feldberg, S.W.InElectroanalytical Chemistry; Bard,A. J., Ed.; Marcel Dekker: New York, 1969 Vol. 3.
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 14,JULY 15, 1993
9
to
10
A Flgure 1. Configurationof long pathlength thin-layer spectroelectrochemicalcell (not to scale). (A) Cell only. (B) Top view of the cell and mount. 1, Teflon plate: 2, polyethylene spacer; 3, working electrode: 4, thin-layer solution: 5, solution inlet and auxiliary electrode; 6, auxillary electrode: 7, reference electrode; 8,screw; 9, optical glass: 10, opaque tape; 1 1 , incldent light beam: 12, gear for adjusting the posffion of the SEC cell.
surface due to considerable iR drop, potential drop due to the uncompensated solution resistance, in the thin-layer solution, the Agi AgCl reference electrode with a saturated KC1 salt bridge was put in intimate contact with the WE, and two platinum wires were placed at two ends of the working electrode to serve as the AE. For the cell for semiinfinite diffusion, the lighttransparent region was defined by four pieces of opaque tape, which were used to define the heights of the front and back light windows just as in the thin-layer cell, the working electrode surface, and the sharp edges of two pieces of blade, which were stuck on the two pieces of glass with 704 silicone rubber adhesive. The two pieces of blade were adjusted carefully to ensure that their sharp edges were located parallel to the WE surface and the width of the thin-layer light-transparent region was uniform, hence, the width was measured simultaneously. The distance between the front light window and the back light window was 1.03cm. The Ag/AgCl reference electrode with a saturated KCl salt bridge was also put in intimate contact with the WE to minimize the iR drop of the solution. A platinum plate with large surface area, which served as the AE, was located parallel to the WE and separated from it by a distance of ca. 3 cm. The alignment of the working electrode with respect to the incident light beam is of obvious importance in experiments and was accomplished as follows. The spectroelectrochemical cell was fixed in a home-made mount with functions of fine rotation and horizontal adjustment. At first, the front light window was located almost always at the center of the incident light beam, then the cell direction was changed by rotating the mount and adjusting the horizontal orientation simultaneously and repeatedly until the greatest transmittance was recorded. This adjustment ensures that the incident light beam passes parallel to the working electrode surface and well through the front and back light windows. Thus, the optical pathlength of the spectroelectrochemical cell used in experiments, 1, was equal t o the distance between the front and back windows. Instrumentation and Reagents. Spectroelectrochemical experiments were carried out with a Hitachi 557 double wave1engtWdoublebeam spectrophotometer and a CSH-2potentiostat (Sanming Electronic Factory). All the potentials were read out from a PHS-3C Model pH-meter (Shanghai Analytical Instrument Factory). All chemicalswere at least analytical grade. Fresh solutions of K4Fe(CN)Bor K3Fe(CN)6were prepared in 0.5 M KCl. Doubly distilled water was used for all preparations. The experiments were done at room temperature. Procedure. The tested solutions were transferred into the spectroelectrochemicalcell carefullyin order to avoid gas bubbles.
The potential-step amplitude was 0.00-0.60 V vs Ag/AgCl/KCl (sat.) for the oxidation process of Fe(CN)& and 0.60-0.00 V vs Ag/AgCl/KCl (sat.) for the reduction process of Fe(CN)s*. After the alignment of the working electrode was accomplished with respect to the incident light beam, the optical signals were recorded in situ after a potential-step perturbation. Computationand Analysis of Data. The digital simulation programs were written in FORTRAN and run on an AST286 microcomputer. In the programs, values of m change with those of w. A greater value for w needs a greater value of m to give accurate results, for example,computation errors can be neglected with m = 40 for w = 0.015 cm and m = 120 for w = 0.04 cm, respectively. Furthermore, for the case of semiinfinite diffusion, the total number of volume elements of concentration distribution has been calculated to the larger one of m and 6(Dot)1/2/(&), where t is the duration of the experiments.
RESULTS AND DISCUSSION Discussion about Theoretical Results. The theoretical absorbance vs time behavior has been obtained by using the methods mentioned above. For case 1, absorbance at first decreases abruptly with time after a potential-step perturbation, then more slowly. The greater the value of t,C,*l is, the faster absorbance decreases with respect to time. The eventual absorbance should be 0 according to eq 5. For case 2, absorbance increases rapidly with time at first and then more slowly. The greater the value of €oCR*l is, the faster absorbance increases with respect to time. And absorbance eventually reaches a steady value ( D ~ D o ) 1 / 2 ~ owhen C ~ *the l solution concentration across the optical beam is uniform and is equal to (DdD,)112cR*according to eq 6. For case 3, absorbance vs time behavior is similar to that for case 2, however, absorbance increases more rapidly with respect to time in relatively long periods of time, and exhausted electrolysis can be achieved more easily, though the two absorbance vs time curves are quite the same in short time when the semiinfinite diffusion condition is obeyed. Absorbance vs the normalized time tD,/w2 for c,C*l = 1.000 and QC*I = 0.000 are shown in Figure 2. Curve 4 presents reduction process of an absorbing, oxidized species under thin-layer diffusion. The initial part in curve 4 is the same as the initial part in curve 1;however, absorbance decreases
ANALYTICAL CHEMISTRY, VOL. 65, NO. 14, JULY 15, 1993
1.0 0.8
I
0.8
.
W
u
rc
0.6
m
I
[kl
8
m
1801
0.4
a
0.2
0.0
I 0
1
I
I
1
0.2 0.4 0.6 0.8
0.0
1
I
d
0
1.0
5
NORMALIZED TIME
10
,
15
28
25
TIME (SEC)
Figwe 2. Absorbance vs normellzed tlme curves when gC'/ = 1.OOO and hRc/ = 0.000. DdL+,= 1-18, 1, case 1; 2, case 2; 3, case 3 4, reduction process under thin-layer dltfuaion; 5, from eq 11.
1.0
Flgwe 4. Absorbance v8 tlme curves for case 1. toCo'/= 0.803, where to and Co* are the molar absorptivity and bulk concentration of Fe(CNk". Values of w (pm)used were as follows: 1,431; 2,322; 3,263; 4,191; 5,147. Circles, experimental results. Lines, theoretical
results.
0.8
w u z
0.6
8
4 ct
0.4
8.6 I
0
c 3
4
8.4 U
0.2
3 2 I
0.2
0.0 0.0
0
0.2 0.4 0.6 0.8
1.0
(tDo) y y w Flguro 3. Al(e,C*I) vs (tDo)1121 wcurves for case 1 and case 2. D&, = 1.18. hRc/ = 0.000. Curves are numbered sequentlally from 1 to 8 from left top to left bottom. Curves 1-4 are for case 1, and the others are for case 2. Computation conditions are as follows: 1, eoCo'/C< 0.1 or from eq 4 with an arbibaty value of gCo*I;2, gCo'/ = 0.200 3, coCo'/= 0.600 4, gCo*l = 1.000; 5, toG'l
