Anal. Chem. 1982, 5 4 , 1727-1729
our previously reported microbial electrode system (9). The sensitivity of the microbial electrode system was higher than the conventional Ames test and our previous system. The minimum measurable mutagen concentration was 0.001 pg mL-l by the microbial electrode, 1.6 pg mL-l by the previous system and 10 fig mL-l by the Ames test for AF-2. The microbial electrode system appears promising and attractive for use in the routine preliminary screening of mutagens and carcinogens. Further developmental studies in this laboratory are beiing directed toward applying immobilized S-9 fraction (microsomefrom rat liver) to the microbial electrode. LITERATURE CITED (1) Bridges, B. A. Nature (London) 1976, 261, 195-200. (2) Ames B. N.; Lee, F. D.; Durston, W. E. Roc. Natl. Acad. Sci. U.S.A. 1973, 70, 782-786.
1727
(3) Ames, B. N.; Durston, W. E.; Yamasakl, E.; Lee F. D. R o c . Natl. Acad. Scl. U.S.A. 1973. 70, 2281-2285. (4) Bridges, B. A.; Mothershead, R. P.; Rothweii, M. A.; Green, M. H. L. Chemdiol. Interact. 1972, 5 , 77-84. (5) Kada, T.; Turikawa, K.; Sadaie, Y. Mutat. Res. 1972, 16 165-174. (6) Karube, I.; Matsunaga, T.; Mitsuda, S.; Suzuki, S. Biotechnol. Bioeng. 1977, 19, 1535-1547. (7) Karube, 1,; Matsunaga, T.; Suzuki, S. Anal. Chlm. Acta 1979, 109, 39-44. (8) Matsunaga, T.; Karube, I.; Nakahara, T.; Suzuki, S. Eur. J . Appl. Microbiol. Biotechnoi. 1981, 12, 97-101. (9) Karube, I.; Matsunaga, T.; Nakahara, T.; Suzuki, S. Anal. Chem. 1981, 53, 1024-1026. (10) Postgate, J. R. “Method in Microbiology”; Academic Press: New York, 1969: Vol. 1. (11) Ames, B. N. Science 1979, 204, 587-593. (12) McCann, J.: Choi, E.; Yamazaki. E.; Ames, B. N. Proc. Natl. Acad. Sci. 1975, 72, 5135-5139.
RECEIVED for review January 4,1982. Accepted May 10,1982.
Theory for Double Potential Step Chronoamperometry, Chronocouloimetry, and Chronoabsorptometry with a Quasi-Reversible Electrode Reaction Dennis H. Evans* and Michael J. Kelly Department of Chemistry, University of Wisconsin-Madison,
Madison, Wisconsin 53706
Serles solutions for the current-, charge-, and absorbancetime responses have been obtained for the case where the flrst step Is to a potential where the forward reaction occurs at a dlffuslon-controlled rate and the second step Is to any potential. A slngle equatlon Is adequate for systems ranging from fully reversible to extremely slow electron transfer reactions. Simple data anallysis can be achieved by using Ilmking forms of the solution for times shortly afler the switchlng time.
The analytical signal from most electroanalytical techniques is affected by the rate of the electrode reaction so it is important to be able to mearriire such rates. A simple approach is to apply a potential step rind measure the current ( I ) , charge (2), or (in transmission mode spectroelectrochemistry) the absorbance (3) as a function of time. For example, the current for reduction of a material 0 is given ( I ) by eq 2 k
o -t.ne .&
R
kb
i = nFACo*kr exp(H2t) erfc kf = k, exp((--anF/RT)(E - E O ) ) kb = k, exp((1 -. a)(nF/RT)(E- E o ) ) H = kf/i901/2 + kb/DR1f2
(1) (2)
(3) (4)
(5) where i is the current, A is the electrode area, CO*is the bulk concentration of reactant (product is assumed to be initially absent, CR*= 0), t is time, k , is the standard heterogeneous electron transfer rate contatant, a is the transfer coefficient, E is the electrode potentkill, Eo is the formal potential, DIis the diffusion coefficient of species I, and the other symbols
carry their normal meaning. The discussion will be developed in terms of an initial reduction step but the results can be easily adjusted to cover the converse. When Ht1I2 is small, a linear plot of i vs. t1/2is obtained which may be extrapolated to t = 0 to evaluate the initial current, itlo. When t = 0, eq 2 becomes
it=,, = nFACo*kf
(6) which provides a simple and direct means of determining kf at various potentials. The direct measurement of k b can be achieved by analogous studies of solutions containing R but no 0. Here the potential steps would be from an initial potential where no oxidation occurs to a more positive potential where the oxidation occurs a t a measurable rate. However, it is frequently infeasible to prepare a solution of R either because it is difficult to prepare in a pure form or because R is unstable. In such cases, a double potential step experiment can be implemented. In the first step, the potential is held at a very negative value where 0 is reduced to R at the diffusion-limited rate. This electrolysis produces a well-defined concentration-distance profiie for the product R. At a switching time, T , the potential is changed to a more positive value where oxidation of R occurs and the observed current can be used to evaluate kb. In an earlier application of this technique ( 4 ) , it was necessary to resort to digital simulation (5) to carry out an analysis of the current-time curve during the second step. Digital simulation was also used to find the best fit with experimental absorbance-time curves in a recent spectroelectrochemical study (6). Clearly, an analytical solution would be very useful. In this paper, we report a series solution which provides a useful theoretical basis for data analysis. Several earlier treatments exist. Smit and Wijnen (7)obtained a solution which is valid only for “symmetric” steps, Le., for cases where H on the first step is numerically equal
0003-2700/82/0354-1727$01.25/0 . 0 1982 American Chemlcai Society
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
to H on the second step. For a = 0.5, this condition requires that the two potentials be on opposite sides of the formal potential and equally displaced from it. A treatment by Bard and Faulkner (8) appears to be more general, and, in fact, it produces the correct reversible limiting form but for quasireversible systems it is only valid for the same condition as that of Smit and Wijnen. Kimmerle and Chevalet (9) derived expressions for the current-time and charge-time responses but the solution is restricted to the case where k b >> k f on the second step. Finally, Aoki, Osteryoung, and Osteryoung (10) have developed a completely general solution as a basis for their theory for differential normal pulse voltammetry. The solution involves two series, one for small Ht1l2and one for large values.
c t
c
I I
THEORY
As the derivation involves application of the normal Laplace transform techniques (11,12), details will not be presented and only the statement of the problem, the general features of the derivation, and the result will be given. Mass transfer by semiinfinite linear diffusion is assumed. Application of the first potential step to a potential quite negative of Eo produces concentrations of 0 and R a t the switching time T given by (13)
CR
rCo*(1- erf (x/22/&7))
(8)
where y = (Do/DR)1/2and x is the distance from the electrode surface. These concentration-distance profiles will be used as initial conditions for derivation of the current during the second potential step. The diffusion equations and boundary conditions are dC,/dt
= Do(d2Co/dx2)
- -
dCR/dt = DR(d2CR/dx2)
t > 7,x
> 7,x t > 7,x t
(9) (10)
cO*; CR+ 0
(11)
= 0: Do(dCo/d~)x=o= -&(dCR/dx),=o
(12)
O0:
CO
= 0: i/nFA = Do(dCo/dx)x,o = kf(CO)x=O- kb(CR)x=O (13)
where k f and k b pertain to the potential on the second step. The initial conditions (7) and (8)are represented in series form (14) using eq 14. Short truncations of this series can provide extremely accurate representations
of Co and CRclose to the electrode surface. In fact, six terms give erf ( 2 ) with only 0.1% error at the point in the diffusion layer where co/co* = 0.9 and cR/ycO* = 0.1. The use of these series versions of the initial conditions generates series solutions for the Laplace transforms of CO and CR. When these are combined with boundary conditions (12) and (13), the Laplace transform of the current is obtained in a form involving two series. The inverse Laplace transforms are then obtained term by term to obtain the solution, eq 15,
i/i7 = -(I
+k
b d x ) exp
~2
erfc
x + 1+
Figure 1. Current-time curves during second potential step for several different potentials: dashed curves, reversible behavior (eq 17); full curves, k , ~ 1 / 2 / D , ' /= 2 1.O by digital simulation; (X), calculated from eq 15. ci = 0.5.
where i, is the current at the end of the first step, X = H ( t - T ) ~ and / ~ a(k) is the coefficient of the ( k + 1)th term of the binomial expansion of (1 ~ ) l / ~ .
+
= [(1/2)(3/2)(5/2)...(k - y2)1/k!
(16)
In Figure 1,values of i/i7 from eq 15 are plotted for five different potentials and ~,(T/DR)~/' = 1. Curves for reversible behavior (k,(T/DR)l/z m) are also shown for comparison. To confirm the accuracy of eq 15, the results ( x ) were compared with digital simulations (smooth curves) and excellent agreement was found. The agreement was found to persist for calculations for the same five potentials over a range of ~ , ( T / D R >from ~ / ~near reversibility (lo4)to values where almost no current exists during the second step (lo4). When performing numerical calculations with small A, the second series in (15) will diverge. The series should be terminated with the value of p producing the term with the minimum absolute value of all terms and this same value of p also should be used to terminate the first series in eq 15. The first series diverges for ( t - T ) / T 2 1 so the solution is limited to values of the time variable less than unity. For ( t - T ) / T 5 0175 rapid convergence is achieved. Limiting Forms. There are two important limiting forms of eq 15. When H d 2 is very large, the second series in (15) is negligible and only the first three terms remain. The remaining series is simply the binomial expansion of [ 1 + ( t T ) / T ] - 1 / 2 and for large values of X, exp X2 erfc A is given by 1/idZA. Under these conditions (15) reduces to (17), which is the equation for the current for the reversible limit where the first step is to a very negative potential and the second is to any potential (15).
-
(17) 0 = exp((nF/RT)(E - EO))
(18)
Equation 17 is an accurate representation of the current when the system is reversible (large k,) or H d Zis large because the potential during the second step is either very negative or very positive. The curves for E = -1.0 and +1.0 V as well as the curves for the reversible responses in Figure 1were plotted using the limiting form, eq 17. The second limiting case is encountered when A is small. In this case the second series in eq 15 is again negligible and
ANALYTICAL CHEMISTRY, VOL. 54, NO. 11, SEPTEMBER 1982
1729
time integral of the current. Integration of eq 15 involves simple integrals of powers of t - 7 and the integral of exp(P(t - 7))erfc ( H ( t - ~ ) l / 7) ~ which ) is (1/H2) x , ,
[exp(H2(t - 7))erfc ( H G )
+ 2Hd-
- 11 (24)
The equation for the absorbance-time response is directly accessible from the charge-time curve via eq 25 A = E&/"rzFA (25) where A is the absorbance, t is the molar absorptivity of R, and Q is the total charge passed since the beginning of the experiment. The limiting version of the chronocoulometriccurve relevant for small H ( t - 7)ll2 is
m t
Q- -Q, QT
Figure 2. Limiting form of current-time curves for times shortly after the switching time: dashed lines, eq 20; full curves, eq 15. k8~1'2/ = 0.1. CY = 0.5.
exp X2 erfc X may be replaced by the first two terms of the series 1- 2X/&
-k A2 - 4X3/3(i
+ ...
(19)
The result is
i/ir = - k b d G
2HC(l/vG
k b d x )
(20) or, expressed as the current itself rather than the current ratio, (20) becomes
which is strikingly similar to the limiting version of eq 2 for the single potential step experiment a t short times
i = nFAkfCo*(l - 2H*)
(22)
Equation 21 predicts that for small values of H ( t - 7)1/2, linear plots of current vs. (t - ,r)1/2will be obtained which may be extrapolated to t = T to evaluate the initial current after application of the second potential step given by := -nFAkbyCo* (23) Thus, the initial rate of itlhe reaction is given by the product of k b and the surface concentration of R at the switching time, yCo* (cf. eq 8). The contribution to the current by reduction of 0 is unimportant because the surface concentration of 0 is zero at T (cf. eq 7). In Figure 2, values of i/il from eq 15 are plotted vs. [ ( tT ) / T ] ~for / ~ ~,(T/DR)'/~ = 0.1 and the results are compared with the limiting form, eq 20. Values of k,(7/&)l/' in the range of 0.05-0.5 are most suitable for data analysis as illustrated in Figure 2. The value of kb is obtained from the [ ( t- T)/T]l/' = 0 intercept. For a given electrode reaction 7 would be selected to bring k,(T/DR)1/2within this range. For example, k, for Fe(CN)83-/Fe(CN:)lj4-a t a tin oxide electrode in pH 7 phosphate buffer is 4.6 >< lo4 cm/s (3) and DR = 6.5 X lo4 cm2/s; thus, T = 0.3 s would be selected to give k,(T/DR)1/2 -0.1.
Double Potential Step Chronocoulometry and Chronoabsorptometry. The chronocoulometric response is the
(26) where Q is the total charge and Q, is the charge passed up to the switching time. Equation 26 is not as convenient as eq 20 but its form suggests data analysis via plots of (Q Q,)/Q,(t - 7) vs. (t - 7)ll2 which should be linear with an intercept of -kb~1/2/2(DR7)1/2. ACKNOWLEDGMENT We thank Henry Blount and Fred Hawkridge for several technical suggestions. Supplementary Material Available: Derivation of equation 15 (9 pages) will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper or microfiche (105 X 148 mm, 24X reduction, negatives) may be obtained from Distribution Office, Books and Journals Division, American Chemical Society, 1155 16th St., N.W., Washington, D.C. 20036. Full bibliographic citation (journal, title of article, author) and prepayment, check or money order for $9.00 for photocopy ($10.50foreign) or $4.00 for microfiche ($5.00 foreign), are required. LITERATURE CITED (1) Gerlscher, H.; Vlelstich, W. 2.Phys. Chem. (FrankfurtlMaln) 1955, 3, 16-33. (2) Christie, J. H.;Lauer, G.; Osteryoung, R. A,; Anson, F. C. Anal. Chem. 1963,35,1979. (3) Albertson, D. E.; Blount, H. N.; Hawkridge, F. M. Anal. Chem. 1979, 51, 558-560. (4) Corrlgan, D. A.; Evans, D. H. J . Electroanal. Chem. 1960, 106, 267-304. (5) Feldberg, S. W. In "Electroanalytical Chemistry"; Bard, A. J., Ed.; Marcel Dekker: New York, 1969;Vol. 3, pp 199-296. (6) Bancroft, E. E.; Blount, H. N.; Hawkridge, F. M. Adv. Chem. Ser., In press.
(7) Smit, W. M.; Wljnen, M. D. R e d . Trav. Chim. Pays-Bas 1960, 79, 5-2 1. (8) Bard, A. J.; Faulkner, L. R. "Electrochemical Methods. Fundamentals and Appllcatlons"; Why: New York, 1980;pp 182-183. (9) Klmmerle, F. M.; Chevalet, J. J . Electroanal. Chem. 1969, 2 1 ,
237-255.
(IO) Aokl, K.; Osteryoung, J.; Osteryoung, R. A. J . Electroanal. Chem. 1980,1 1 0 , 1-18.
(1 1) MacDonald, D. D. "Transient Technlques In Electrochemistry";Plenum Press: New York, 1977;Chapter 3. (12) Bard, A. J.; Faulkner, L. R. "Electrochemlcal Methods, Fundamentals and Appllcatlons"; Wlley: New York, 1980;pp 657-865. (13) Bard, A. J.; Faulkner, L. A. "Electrochemlcal Methods, Fundamentals and Applications"; Wlley: New York. 1980;p 161. (14) Abramowltz, M., Stegun, I. A., Eds. Handbook of Mathematical Functlons"; Dover: New York, 1964,p 297. (15) Kambara. T. Bull. Chem. SOC.Jpn. 1954,27, 527-529.
RECEIVED for review April 12,1982. Accepted June 3,1982. This research was supported by the National Science Foundation, Grant No. CHE 81-11421.
