1006
Anal. Chem. 1981, 53, 1006-1011 Erlksson. Karl-Erik: Marie-Claude: Krlnastad. Knut Sven. PaDDefStldn. .. 1979, 82, 95-104. Bjorseth, Alf; Carlberg, Georg E.; Moiler, Mona. Sci. Total Envlron. 1879 1 1 . 197-211
Rappapoi, Stephen M.; Richard, Michael G.; Hollsteln, Monica C.; Talcott, Ronald E. Environ. Sci. Techno/.1979, 13, 957-961. McCann, Joyce; Chol, Edmund; Yamasakl, Edith, Ames, Bruce N. Proc. Natl. Acad. Scl. U . S . A l 9 7 5 , 72, 5135-5139. Miller, Elizabeth C.; Miller, James A. In “Chemical Carcinogens”; Searle, Charles E., Ed.; American Chemical Society: Washington, DC, 1976; pp 737-762. Epsteln, Joseph: Rosenthal. Robert W.; Ess, Richard J. Anal. Chem. 1955, 27, 1435-1439. Friedman, Orrle M.; Boger, Ellahu Anal. Chem. 1961, 33, 906-910. Sawlckl, Eugene; Bender, D. F.; Hauser, T. R.; Wilson, R. M., Jr.; Meeker, J. E. Anal. Chem. 1963, 35, 1479-1486. Sladek, N. E. Cancer Res. 1971, 37, 901-906. Barbin, A.; Bresll, H.; Crolsy, A.; Jacqulgnon P.; Malavellle, C.; Montesano, R.; Bartsch, H. Biochem. Biophys. Res. Commun. 1975, 67, 596-603. Asalmi, Masato; Nakamara, Kan-lchl; Kawada, Katsuro; Tanaka, Minoru J . Chromatogr. 1979, 174, 216-220. Agarwal, Satlsh C.; Van Duuren, Benjamin L.; Knelp, Theodore J. Bull. Environ. Confam. Toxlcol. 1979, 23, 825-829. March, Jerry “Advanced Organic Chemistry”; McGraw-HIII: New York, 1968; p 290. Kline, Stanley A.; Van Duuren, BenJamln L. J. Heterocycl. Chem. 1977, 14, 455-456.
(23) Barnett, Ronald E.; Jencks, William P. J. Am. Chem. SOC. 1969, 97, 6758-6765. (24) Cheh, Albert M.: SkochdoDole, Jill: Koskl, Paul: Cole, Larry Sclence 1960, 207, 90-92. (25) Rosenkranz, Herbert S.; Speck, William T. Blochem. Blophys. Res. Commun. 1975, 66, 520-525. (26) Ames, B. N.; McCann, J.; Yamasakl, E. Mutat. Res. 1975, 31, 347-352
(27) McCaii: Joyce; Ames, Bruce N. Proc. Natl. Acad. Sci. U.S.A. 1976 73, 950-954. (26) Cheh, Albert M.; Skochdopole, Jill; Heillg, Charles; Koskl, Paul M.; Cole, Larry In “Water Chlorination: Environmental Impact and Health Effects”; Jolley, Robert L., Brungs, W. E., Cummlng, R. 8.; Eds.; Ann Arbor Science: Ann Arbor, MI. 1960; pp 603-615. (29) Price, C. C.; Stacy, G. W. J . Am. Chem. Soc., 1946, 68, 498-500. (30) Hemmlnkl, K.; Falck, K. Toxlcol. Lett. 1979, 4 , 103-106. (31) Bartsch Helmut; Malavellle, Chrlstlan; Barbln, Alaln; Planche, Qhyslalne Arch. Toxicol. 197% 41, 249-277.
RECEIVED for review September 26,1980. Accepted February 4,1981. This work was supported by funds from the Freshwater Biological Research Foundation. It was presented in part at the Eleventh Annual Meeting of the Environmental Mutagen Society, Nashville, TN, March 16-19, 1980.
Determination of Rate Constant of Quasi-Reversible Electrode Reaction by Mechanical Square Wave Polarography Feng Qiang-sheng” and Lin Sin-rhu Shanghai Institute of Metallurgy, Academia Sinica, Shanghai 200050, The Peoples’ Republic of Chlna
An approximate equation for the quasi-reverstble electrode reactions of DME square wave polarography has been obtained. A simple and convenlent method for determining the rate constants of fast electrode reactlons by mechanical square wave polarography Is presented, in which the reslstance of the system Is reduced so that rate constants close to 1 cm/s can be determined. The use of anaiysls in Laplace space for square wave polarography Is described.
Barker (1) and Matsuda (2) derived the equation of the quasi-reversible electrode reaction for square wave polarography. Articles (3-6) which use square wave polarography to study the determination of the kinetic parameters of fast electrode reactions were not found in the literature until the last few years. Besides the difficulty with mathematics, the reason square wave polarography has only recently been used for studying fast electrode reactions is that the resistance of the amplifier recording current must be large in order to gain higher sensitivity; thus the time constant is too large to study fast electrode reactions. If the resistance of the amplifier is reduced, the noise of the amplifier will be increased. If the concentration of depolarizer is increased to increase the signal-to-noise ratio, the determination will be affected by the occurrence of polarographic maxima. Square wave polarography is useful for determiningthe rate constants of electrode reactions, whose experimental upper limit of determination was thought to be 0.1 cm/s (3,7). Practically, however, as pointed out by Heyrovsky (7): “So far ...the theoretical calculations have not been tested.” The aim of this paper is to make use of mechanical square wave polarography in the field of fast electrode reactions by decreasing the resistance of the whole system and setting up a simple and convenient method for determining the rate
constants of fast electrode reactions. Mechanical square wave polarography (8)has very high sensitivity and can record the response of current at any elapsed time since the last change in square-wave sign. So it is a suitable method for studying fast electrode reactions. One advantage of this instrument is that the resistance of the whole instrument is close to zero while eliminating the charging current in a half period of every square wave. In addition, we designed a capillary whose resistance is less than 1Q. Thus, the resistance of the whole system is quite small: the time constant can be less than 5 p. The upper limit of the rate constants determined was over 0.1 cm/s. On the basis of Barker’s equation for a plane electrode ( I ) , we obtained a simple, approximate equation for the DME, and to avoid complicated calculations, we used analysis in Laplace space to handle the data.
THEORY Consider a quasi-reversible electrode reaction at the DME
Ox + ne
Red kb
where kf and kb are heterogeneous rate constants of the electrode reactions: k f characterizes the reduction reaction and kb the oxidation reaction. It is assumed that reduction of Ox to Red is a first-order process. No reduced form is present in the electrolyte before electrolysis. Ox approaches the electrode surface as a result of linear diffusion. Red formed in the electrode reaction is soluble either in the solution or in the Hg phase, and it diffuses away from the electrode surface. Koutecw (9) obtained an exact solution for a potential step in the above process
0003-2700/61/0353-1006$01.25/00 1981 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
1007
Table I. Results of Comparison of Equations 5 and 6 with Equation 2 %
X
F(X)
F,(X)a
errorb
%
F,(X)c
errord
0.005 0.01 0.05 0.1 0.2 0.4 0.6 0.8 1
0.004 41 0.004 419 t 0.2 0.006 740 t 52.83 0.008 80 0.008 813 t0.15 0.013 442 +52.75 0.042 83 0.043 09 t0.65 0.064 866 +51.52 +l.27 0.124 453 t48.44 0.082 79 0.083 84 0.155 1 0.158 89 +2.44 0.229 740 t44.59 0.274 9 0.286 79 t 4 . 3 3 0.395 738 t 4 4 0.368 8 0.391 659 1-6.2 0.517 991 t40.45 0.444 0 0.475 578 t 7 . 1 1 0.609 631 +36.7 0.605 0 0.545 642 t 8.05 0.679 466 t34.55 3 0.773 0.854 873 t 1 0 0.923 728 t19.5 5 0.857 7 0.941 11 t9.72 a F,(X) = 7 ~ ' / ~ ( Xe/ ~ 2p ) ( X l 2 )erfc(X/2). ~ F,(X)/ F,(X) = i ~ l / ~ u texp(ut1/2)2 "~ erfc(utllz). F(X). F,(X)/F(X).
T
Figure 1.
Schematic representation of square wave.
where i is the current clue to fast electrode process and i, is the current when the rate of electrode process was only transport controlled, and
x = 2fiut'J2
(3)
(4)
-
I
\
if the process is irreversible, kb 0, u = kf/Do1l2. Delahay (IO),who took into account the increase of current due to the increase in area of the DME, multiplied the result for the plane electrode by (7/3)'12
i = n F q C o b k f f l exp(ut1/2)2erfc (ut1f2) (5) where Cob is concentration of Ox in the bulk solution. u = kf/Do1/2.The other symbols are used in their usual significances. Kouteckys equation is much too complicated to determine the i, at the same time the i is determined, and the F ( X ) is too complicated to be of direct use for square wave polarography. The accuracy of Delahay's equation is poor, the largest error being >+50%, as compared to Kouteckys equation. We resolved the equation for the process ( I ) and introduced an approximate expression
i= n&(k&ob - kbCRb) e:Kp(@ut1/2)2
erfc (@Ut'/')
(6) This is the same form iBs Randles' (11) empirical equation which was obtained by putting the (3/7)lI2 into the result for the stationary plane electrode. As compared to Koutecky's equation the error of eq 6 is 1+10%. For results of comparison of eq 5 and 6 with eq 2, see Table I. Thus, according to the principle of superposition, we derive the relationship between the peak current of square wave polarography and time from eq 6
x i, = nFq(kfCOs- kbCRE') m=O
exp[M(m
+
r)lI2l2erfc [M(m + r)1/2] (7) where q (cm2) = the surface of the DME, Cs(M/cm3) = the concentration of substance a t the surface of the electrode, m = the amount of half cycle of square wave, and r = t / r where 7 (s) = the half-period of square wave (see Figure 1)and t (9) = the elapsed time since the last change in square-wave sign
M=2
(8)
Figure 2. Capillary with lower resistance: (1) glass capillary, (2) platinum wire.
DO (cm2/s) = the diffusion coefficient of Ox, DR (cm2/s) = the diffusion coefficient of Red, and kl12 (cm/s) = the rate constant of the forward electrode process of eq 1(the reduction of Ox) at E = E , (peak potential). Equation 7 is the same form obtained by Barker ( I ) . But, eq 7 is suitable for DME. It follows from eq 8 that if M is known, the value of the rate constant can be calculated out
EXPERIMENTAL SECTION Apparatus. A mechanical square wave polarograph (Model 895, made in The Peoples' Republic of China) was used in our experiments. Its simple description can be read in Appendix 1. The peak to peak amplitude of the square-wavevoltage was 0.005 V, the frequency of square wave was 44 Hz. The capillary (see Figure 2) with lower resistance was constructed by inserting a platinum wire (4 = 0.1 mm) into a usual capillary (4 = 0.134 mm). The flow rate was 0.71 mg/s, and the natural drop time was 13.1 s in distilled water. The counterelectrode was a Hg pool. Reagents. The solution was prepared with high-purity metals, analytical reagents, and redistilled water; the dissolved oxygen was expelled by bubbling argon through the solution before electrolysis. All experiments were performed at 25 i 0.2 "C. The calculations were programmed on a Model 719 computer (made in The Peoples' Republic of China). RESULTS AND DISCUSSION The Upper Limit of the Concentration of Depolarizer. The determination of peak current will be affected by the occurrence of polarographic maxima. Under identical solution conditions, if the ratio of i, (peak current) to id (diffusion current) determined with and without maximum surpressor , m is unchanged, it may be said that no maximum is present.
1008
ANALYTICAL CHEMISTRY, VOL,
53, NO. 7, JUNE 1981
rp
Table 11. & Values at Various Concentration of Pb*", Cdl", and Zna+ Pb2+in 0.5 Cd2+in 0.5 Zn2+in 0.5 M KCl M Na,SO, M KCl -. concn, M a b a b a b 1.0 X l o p 3 2.35 2.19 2.00 2.04 0.89 0.74 5.0 X 2.20 2.15 0.81 0.67 2.0 X 1.97 2.04 0.64 0.67 1.0 X lo-, 1.95 1.93 2.04 2.00 0.70 0.71 5.0 X l o - ' 2.08 2.00 0.61 0.67 a N o Triton added, 0.001% Triton added.
Figure 3.
Polarograms of Cd2+ (1.9 X
solution.
M) in 0.5 M
Na2S04
Table 111. A(36r, = 1, 36r, = 31)Values at Various M Values M
10 9 8 7 6 5 4.5 4 3.5 3 2.8 2.6 2.4 2.2 2 1.8
A(36r, = 1, 36r2 = 31) 6.602 6.485 6.327 6.106 5.813 5.423 5.180 4.904 4.582 4.21 8 4.058 3.863 3.690 3.528 3.334 3.127
M
1.6 1.5 1.4 1.3 1.2 1.1 1
0.9 0.8
0.7 0.6 0.5 0.4 0.3 0.2 0.1
A(36r, = 1, 36r2 = 31) 2.914 2.804 2.692 2.5 57 2.460 2.342 2.223 2.099 1.957 1.850 1.720 1.603 1.472 1.346 1.221 1.100
Flgure 4. Comparison of the experimental data of Cd2+with theoretical
values.
-- i,(r2)
K
exp[M(m
m=O
+ r1)'l2I2 erfc[M(m + r1)1/2]
K
-
C (-I)~ exp[M(m + rg)1/2]2erfc[M(m + r ~ l / ~ I
m=O
&1,r2)
30
36 Yt
It follows from the experimental results (see Table 11)that M, no when the concentration of depolarizer is 1 2 X maxima are present in general. Determination of k l l zValues. It can be seen from eq 7 that the ratio of two peak currents obtained at the different elapsed time since the last change in square-wavesign can be exprdssed as ip(rl)
C
I 20
/O
(10)
If the A(rl,r2) values at various M were calculated out by computer and tabulated (see Table 111) in advance, having consulted the M value an the bask of the ratio of the experimental values of two peak heights, the kllz can be calculated according to eq 9. From the following example, it is evident that theory is in accordance with experlmehtal results: The comparison of the experimental data of Cd2+(1.9 X M) in 0.5 M Na2SO4 with theoretical values is shown in Figures 3 and 4.
In Figure 4 the dark squares represent the experimental values of ip(r1)/ip(36r2= 31) obtained at various rl. The curve represents the theoretical values of A(r1,36r2= 31) at M = 3.5 (from Table I11 according to the experimental values of ip(36rl = l)/ip(36rz= 31) = 4-59). It follows from Figure 4 that the experimental values coincide with the theoretical values and that the method for calculating k l l z values is reliable. Some element kllz values, evaluated from experimental data, can be seen in Table IV. It is obvious that the values in the Table IV are in fair agreement with the values obtained by other investigators (1, 12-14). It can be known from Table IV that the k l l z value of Pb2+ in 0.5 M KC1 is 2.3 X 10-1 cm/s. Thus it may be said that the electrode reaction of Pb2+is close to reversible. The ip-t experiniental curve of Pb2+ and the theoretical curve of a reversible process are shown in Figure 5. From the above results of experiments we can see that the upper limit of rate constant of electrode reactions determined may be close to 1cm/s. If the frequency of the square wave is increaled, the upper limit of rate constants determined will be raised. The value of K is dependent on the drop time (at a certain frequency of square wave). When K > 100 at a certain M value, the difference between A(rl,rz) values is very little. For example, when M = 1, the A(36rl = 1, 36r2 = 31) values at K = 100,200, and 400 are 2.165,2.196, and 2.221, respectively. Thus, when the drop time is 2 3 s, the data tabulated in Table 111 can be used.
Table IV. Values of h , , , at 25 "C electrolyte (concn, M ) cation Pb2+( 2 X l o v 4 ) CdZ+(1.9 x
1n3+(1.1 x 10-4) 1n3+(1.1 x 10-4) 1n3+ (1.1x 10-4)
salt KC1 (0.5) Na,SO, (0.5) KBr (1) KCl(1) KCl(O.5), KNO, (0.5)
electrode reaction Pb2+3- 2e- Pb(Hg) Cd2+t. 2e- + Cd(Hg) In3++ 3e- In(Hg)
* *
-_____
k , , , , cmls
2.3 X 7.2 X 9.4 x 7.9 x 1.9 x
lo-' IO-' 1010'2
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE
r
1
I
1000
1981
Table V. W Values at Various M and r ( K = 400)
W values at 36r = M
1
6
10 2 0.1
1.12 1.25 1.51
1.29 1.37 1.52
33 1.58 1.59 1.53
18 1.47 1.51 1.53
Table VI. Mean Values of W at Various M ( K = 400)
-
M
wm
M
@m
10-2.6 2.6-2.0 8.0-1.3 1.3-1.0
1.25
1.0-0.8 0.8-0.5 0.5-0.1
1.45 1.5 1.55
1.3
1.35 1.4
Table VII. Experimental Dataa of M Pbl+ in 0.5 M KCl 2X 20
IO
36r
30
&io-'
36):
Figure 5. Solid curve: iheoreticai curve of a reversible process. Dashed curve: experimental curve of Pb2+ (2 X I O 4 M) in 0.5 M KCI.
Practically, the current cannot be recorded at a certain transient, and yet muet be during an interval. Thus, what was recorded is rather electric charge than current. The expressions of the relatijon between charge and time have been obtained, they can be read in Appendixes 2 (for plane electrode) and 3 (for DME). The N(rl,r2) values are analogous to A(rl, r2)values in Table 111. The shorter the interval is, the less the difference between both will be. T h e Application of Analysis in Laplace Space to Square Wave Polarography. Macdonald (15) pointed out that using analysis in Laplace space to evaluate the rate constant can elimihatethe necessity for heavy and complicated calculations. However, i t is difficult to directly use this method in square wave polarography, as the current is a summation of many potential steps ( E = E,). Having introduced a W value, the analysis in Laplace space becomes suitable to square wave polarography. The results obtained by us are as follows: If an electrode reaction is quasi-reversible, the concentration of substance reduced or oxidized, Cs,can be approximately calculated out by Ilkovic's equation (16)
Thus, the relation between peak current, i, with time may be written as
. n2F K zp = F T q A E C o b k l p :C (-1)"' m=O
exp[M(m
io
5 15.8 1.65
31.6 1.19
i, P A a Q = 0.00918 cm2, D = 10 X Cob = 2 x lo-' M/cm3.
16 60.6 0.88
31 98 0.55
cm2/s,E = 0.005 V,
Table VIII. Values of N(r,r') at Various M Values (36r= 1, 36r' = 31) M
N
M
N
5.080 6.335 6 6.318 5.5 4.938 6.280 5 4.782 6.176 4.5 4.596 4.379 10 5.732 4 9.5 5.679 3.5 4.133 9 5.622 3 3.832 8.5 5.558 2.9 3.777 8 5.485 2.8 3.708 7.5 5.404 2.7 3.647 7 5.310 2.6 3.570 6.5 5.203 2.5 3.494 a K = 400, r2 - rl = r2' - r,'
50 40 30 20
M
N
2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5
3.426 3.344 3.274 3.184 3.106 3.011 2.934 2.851 2.748 2.650 1.4 2.558 1.3 2.450 = 1/36.
M
N
1.2
2.937
1.1 2.256 1 2.147
0.9 0.8 0.7 0.6 0.5
0.4 0.3 0.2 0.1
2.015 1.928 1.796 1.706 1.581 1.470 1.363 1.273
as above, there is a mean value Wmat a certain M value. Table VI shows these values. From eq 12
i(t) = ip(t)W
(13)
thus a linear equation was obtained from eq 11,12, and 13, 1 = A BS'/~ L [ip(t)]s1/2
+
+
r)1/2]2 erfc[M(m
3 9.48 2.24
1
3.16 3.6
s
+ r)l/2] (11)
If the time of recording current is the same, the difference between the peak current of square wave polarography and the current of an ideal single square wave potential step ( E = EP)may be expressed as
where L is Laplace transform operative symbol, s is a positive real number, and
A=
4RTW (for plane electrode) (15) n2FqALXobDo1/2
A=
4(3/7j1/2RTW n 2 p qAECobDoll
(for DME)
(15a)
exp(Mr1/2)2erfc(Mr1/2) K
C (-11~ exp[M(m + r)1/2]2erfc[M(m + r)ll2]
(12)
m=O
It is obvious that W is dependent on M and t (at certain M values), it has been determined by calculation that the W value changes from 1.2 to 1.6 M changes from 10 to 0.1 (see Table V). We may take an mean value of W , W = 1.4. The same
Quadratic approximation is suitable here. We have obtained the kll2 value of Pb2+(2 X M) in 0.5 M KCl solution by means of analysis in Laplaee space. The experimental data and result are in Table VII. From these data, b = 2RT/n2FqAECob = 0.0145, make the plot of
1010
ANALYTICAL CHEMISTRY, VOL. 53, NO. 7, JUNE 1981
Table IX. Values of N at Various K Values (M 5 0 ) K 200 400 600 800 1000 N(36r = 1, 6.3341 6.3352 6.3371 6.3384 6.3395 36r' = 31)
1
where C means electric charge and I-'is the Laplace inverse transform operative symbol, the following equation was obtained
where
bears a resemblance to A(rl, r2), we got N(rl, rl')
3. Square Wave Polarography. DME. Expression of C - t . According to C(t2, t l ) = 12 i dt
+
q = 0.85fL/3T2/3= 0.85f/3(m~ t)2/3
(22)
(where f is flow rate, the significances of T and t can be known in Figure l),thus, from eq 7 icms)
Figure 6. Plot of 7pe-s'vs. t .
t ) 2 / 3exp[ 1u2(rnr+ t ) which may be written as K
C = 0.i5nFf2/3(kfCOs- khCRs)C (-l)m X I
,
I
20
15
I
Sf
Figure 7. Plot of
l l L ~ p ( t ) ] S 1vs. ' 2 S'/*.
Let
zpeT;Stagainst t. The area under the curve means the value of L[z,(t)] (see Figure 6). Then make the plot of l/L[z,,(t)]~'/~ against s1j2(see Figure 7). From Figure 7, we know the slope B 0.08. Thus, k i p = bW/B = (0.0145 X 1.4)/0.08 = 0.25. From eq 8 we obtain M = 10 and from Table VI, we know = 1.25. We finally obtain
rm
This is the same value shown in Table IV. It follows from above that handling experimental data by analysis in Laplace space will bring a satisfactory result.
APPENDIX 1. Some Details Regarding the Mechanical Square Wave Polarograph (Model 895). A mechanical square wave polarograph consists of an ordinary polarograph and alternator. The alternator is constituted with four sets of pure gold contact points (noise < 0.01 pV) driven by a motor. The contact frequency of these points depends on the rotating velocity of motor. The rotating velocity is 44,22, and 11Hz in our instrument. The length of contact time of these points can be altered by adjusting the nut. The phase of contact time between A, B sets and C, D sets can also be adjusted by modulating the relative position between the two eccentric disks. For more detailed information regarding the circuit for recording the variation of i vs. time, refer to original literature (8, 17). 2. Square Wave Polarography. Plane Electrode. Expression of C - t . After Laplace transformation of eq 6, according to
F(Tm) =
sT*
TI Fl3exp( l u 2 T ) erfc( 1u2T)'"
dT (24)
then
Equation 24 was handled by inverse Laplace transformation followed by intergrating, which then yielded
+
(mT t1)(10+3#)/e]r ( 'Z1+l)
(26)
The expression of C-t for DME was obtained from eq 25 and eq 26.
ACKNOWLEDGMENT The authors thank Tai Jia-Xing for his assistance. LITERATURE CITED Barker, G. C. Anal. Chlm. Acta 1958, 18, 118. Matsuda, H. Z . Nektrochem. 1958, 62, 977. Tamamushi, R.; Matsuda, K. J . Electroanal. Chem. 1977, 80, 201. Jindai, H. L.; Matsuda, K.; Tamamushi, R. J. Nectroanal: Chem. 1978, 90, 185. Tamaushl, R. J . Electroanal. Chem. 1978, 90, 197. Matsuda, K.; Tamamushi, R. J . Electroanal. Chem. 1979, 100, 831. Heyrovsky, J.; Kuta, J. "Principles of Polarography"; Publishing House of the Czechoslovak Academy of Sciences: Prague, 1965; pp 489. Feng, Q. S.; Liu. 0. L. HuaXue XuePao 1965, 3 1 , 291. Kouteckp, J. Collect. Czech. Chem. Commun. 1953, 18, 597.
1011
Anal. Chem. 1981, 5 3 , 1011-1016
Delahay, P. J. Am. Chem. SOC.1953, 75,1430. Randles, J. E. 5. Can. J . Chem. 1959, 37,238. Gerischer. H. Z. E/ektrochem. 1953, 57,604. Timmer, B.; Sluyters-Rehbach, M.; Sluyters, L. H. J. flectroanal. Chem. 1968. 19, 85. Sluyters-Rehbach, M.; Breukd, J. S. M. C.; Sluyters, L. H. J . Nectroanal. Chem. 1968, 19, 73. Macdonald, D. D. “Transient Techniques in Electrochemistry”; Plenum: New York, 1977; pp 132.
(16) Barker, G. C.; Faircloth, R. L.; Gardner, A. W. U.K., At. Energy Res. €stab/., CIR 1958, 1786. (17) Feng, Q. S. HuaXue XuePao 1966, 32,7.
RECEIVED for review October 26, 1980. Accepted February 23, lg81* This paper was presented at the 180th Meeting Of the American Chemical Society, San Francisco, CA, Aug 1980.
Explicit Finite Difference Method in Simulating Electrode Processes Renato Seeber *
Istituto di Chimlca Generale deli’Universit5 di Siena, Piano dei Mantellini, 44,
53 100 Siena, Italy
Stefan0 Stefani
Istituto di Matematica de8’UniversitS di Siena,
Via del Capltano, 15, 53100 Siena, Italy
The expllclt flnite dlfference method is widely used In slmulating electroanalytical experiments; however, In many cases the required computation amount becomes too large. Two means of overcomlng Nhls dlfflculty are descrlbed In this paper. In the first, a nonunlform space-tlme grld ls built up. If the slre of space elements Increases at Increasing dlstance from the electrode according to a geometrlcal progression, It Is posslble to achieve also a nonunlform tlme dlscretlzatlon, such that the species concentratlons are modlfled wlth a varlable frequency decreaslng at Increasing distance from the electrode. I n the second, the transient behavior of the electrode boundary Is improved leading to accurate values in the computed response even at the early polnts after a sudden change In equilibrium conditions. Efficlency and accuracy of the model are tesUed in simulating different electrode mechanlsms wlth different electroanalytical techniques.
In the last 15 years the digital simulation technique has become a powerful tool in elucidating electrode mechanisms. Its usefulness has been proved in the prediction of the theoretical trend of the significant parameters relative to responses obtained with different electroanalytical techniques and in establishing the nature of an electrode process by comparison between experimental and computed response. The simulation techniques are very useful in solving problems which invollve equations with time-dependent boundary conditions or nonlinear differential equations and, in general, in the cases in which a complete analytical solution becomes either difficult or unfeasible. Among the different possible methods which can be followed in the digital simulation, the explicit finite difference method still retains full validity, as the implicit finite difference method (1-4)seems to be less suitable to solve the boundary value problem if the involved partial differential equations are nonlinear. Although the model involving uniform space and time discretization (5-7) is still satisfactory, also because of its simplicity, perhaps too little attention has been devoted to search possible improvements on it. In particular, some problems arise in the case of the occurrence of homogeneous chemical reactions which are so fast that diffusion layer 0003-2700/81/0353-1011$01.25/0
thickness results much greater than reaction layer. Together with some improving analyses of the classical model (8)we find in literature few examples of valuable suggestions to reduce the amount of computation, introducing a nonuniform space discretization (3) and treating the homogeneous chemical reactions coupled to the charge transfer in an unconventional way (9). In view of these facts the present paper reexamines the classical “Feldberg’s model”, reconsidering both space and time discretization, and the expression of the boundary conditions for the electrode kinetics.
SPACE-TIME DISCRETIZATION In an electrochemical experiment the diffusing perturbation arises at the electrode surface; the concentrationsof the species involved in the electrode process may show abrupt changes near the electrode, while the concentration profiles appear always smoother, increasing the distance from it. This suggests that by considering a nonuniform space discretization the concentrationprofies are still suitably described (3). A similar approach allows the use of a very close-mesh space grid near the electrode, as is for instance required by the occurrence of fast homogeneous chemical reactions coupled to the charge transfer, without a prohibitive increase of the computation amount. One finite-difference approximation of the differential equation for the planar diffusion
following the explicit method ( 1 0 , l l )is, in the case of uniform space discretization, a set of linear equations
+
= P(Ai-lj - 2Aij A i t l j ) (2) where Aij is the concentration value at the ith space element and at the j t h time increment and /3 is the so-called “dimensionless diffusion coefficient”. However, if the space discretization is nonuniform, for every space element two coefficients have to be considered, so that eq 2 becomes A i j t l - Aij = Pr,i(Ai+lj- Aij) + Pl,i(Ai-lj - Ai;) (3) where P,i and are the “right diffusion coefficient” and “left diffusion coefficient”, respectively. A possible way to evaluate Aijtl
- Ai,
0 1981 American Chemical Society
