Theory of Potential-Step Transmission Chronoabsorptometry Chia-yu Lil and George S. Wilson2 Department of Chemistry, University of Arizona. Tucson, Ariz. 85727
This report presents correlations of previously published theory of chronoabsorptometry after a potential-step perturbation. It has been extended to include CE (reversible chemical reaction preceding charge transfer), EC (reversible chemical reaction following charge transfer). reversible catalytic, and disproportionation reactions. The standard Laplace transform technique was used whenever possible to obtain analytical solutions. For the cases in which no analytical solution is available, the digital simulation method was used instead to construct working curves. Quantitative diagnostic criteria based on the characteristics of each absorbing species were developed to permit the assignment of mechanisms for unknown systems with reasonable certainty. Emphasis is placed on using the total absorbance change to determine pertinent parameters.
The development of the optically transparent electrode (OTE) (1-3) has greatly facilitated the study of electrochemically generated intermediates and their associated products. By monitoring the course of the electrochemical reactions by transmission (4-8) or internal reflection (3, 9) techniques, it is possible to obtain not only kinetic parameters but spectral characteristics of reactive intermediates as well ( I O ) . This latter objective is greatly facilitated by the development of the rapid scan spectrophotometer ( 1I ) . Kuwana and coworkers have demonstrated the applicability of potential-step chronoabsorptometry to the determination of reaction rate constants. C'sing working curves, they have studied the cases involving simple charge transfer (j),two successive charge transfer reactions (EE) ( 7 ) , irreversible homogeneous chemical reaction following sin-
Present address, Department of Chemistry, East Carolina University. Greenville, K.C. 27834. To whom communications concerning this paper should be addressed. Kuwana, R . K . Darlington, and D. W . Leedy, Anal. Chem.. 36, 2023 (1964) R . W. Murray, W. R. Heineman, and G . W . O'Dom, Anal. Chem.. T.
39, 1666 (1967). H . 6 . Mark, J r . , L . Winstrom, J. Mattson, and B. Pons, Ana/. Chem.. 39,685 (1967). T. Kuwana and J. W . Strojek, Discuss. faraday SOC.. 45, 134 (1968). J. W. Strojek. T. Kuwana, and S. W . Feldberg, J . Amer. Chem. Soc.. 90, 1353 (1968). H . N . Blount. N. Winograd and T. Kuwana, J. Phys. Chem.. 7 4 , 3231 (1970). G. A. Gruver and T. Kuwana, J. Electroanal, Chem.. 36, 85 (1972). N . Winograd, H . N. Blount, and T. Kuwana, J. Phys. Chem.. 73, 3456 (1969). N. Winograd and T. Kuwana. J. Amer. Chem. Soc.. 93, 4343 (1971). B. P. Neri and G .S. Wilson, A n n . N . Y . Acad Sci.. in press. J . W. Strojek, G . A . Gruver. and T. Kuwana, Anal. Chem.. 41, 481 (1969).
2370
gle charge transfer (EC) ( I 2 ) , homogeneous chemical reaction following two charge transfers (EEC) (7), and first- (8) as well as second-order catalytic reactions (6). The majority of previously studied systems have involved spectral monitoring a t a single wavelength under conditions where only one of the species present absorbs. However, in order to obtain spectra of reactive intermediates, it will be necessary to interpret time-dependent spectral changes by knowing the spectral contributions of each of the species present. To this end, the present report gives the time dependent absorbances of all species encountered in several typical reaction sequences. Where possible, an analytical solution to the boundary value problem was obtained using standard Laplace transform techniques. In some cases such as those involving higher order chemical reactions inverse transforms cannot be found. In such cases, the explicit finite difference (digital simulation) method of Feldberg (13) was then used to establish working curves. Kuwana and coworkers have developed a generalized digital simulation program which can be readily adapted to accommodate various kinetic complications (14).
EXPERIMENTAL All solutions derived by both rigorous and simulation methods were obtained by assuming: a ) semi-infinite linear diffusion during the entire course of a reaction, b) planar electrode geometry (single surface), c) equal diffusion coefficients for both reactants and products except Cases I and I1 in which this assumption is not necessary. d ) a potential step from an initial value a t which no significant electrode reaction occurs to a final value a t which t h e concentration of electroactive species a t the electrode surface is zero (or more practically 0.1% of the bulk concentration, and e ) no correction for double layer charging. All calculations were performed on a Hewlett-Packard 2100A computer (16-K core memory) equipped with a 2.5-million word disk memory. For simulation computation, one hundred calculation loops for one d a t a point (within 1% accuracy) are usually enough. This is roughly equivalent t o 30 seconds computing time with output printed on a high speed H P 2767A printer (600 lines/ minute). (For large arrays, the disk can be used to extend the effective core size by an additional 64 K.) The length of execution time, of course, depends on t h e complexity of diffusion equations, the number of volume elements usually 6 m , the number of iteration loops and the efficiency of the programming. For long time or large rate constant simulation, it is not unusual to employ 1000 or more iteration loops, thus taking much greater computing time.
THEORY The results of the mathematical derivations are presented in Table I. Previously reported solutions are also included. (12) G. C. Grant and T. (13) S. W. Feldberg in (14)
Kuwana. J. Electroanai. Chem.. 24, 11 (1970). "Electroanalytical Chemistry," Voi. 3, A. J. Bard, Ed., Marcel Dekker. New York, N . Y . , 1969, p 199. W. R. Heineman and T. Kuwana. Unpublished results, 1972.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 14, DECEMBER 1973
Table I . Potential-Step Chronoabsorptometric Relations Reaction type
Case I Afne-B
Case I I A f n e T B Z+B-C
Case I I I Afne-B ki
Z + B T C
Case IV kr
Z-A
kb
Afne-B
kb
>> k f
(or K
IO0
( K = kfCz/kt,; for Case I I refer to Curve E only)
'/2
Figure 4. Effect of rate constants on time-dependent absorbance response. Reversible EC mechanism (Case I l l ) for K = 1 . For species B: A l v d -= A g V d = A s ( X , t ) / ( 2 t g ( X ) C ~ * ~ / v ~ ) ; For species C: = A c z V f i= A c ( X , t ) / ( 2 t c ( X ) C ~ * v a / fi). - for species B. ---for species C. Curves A, A'
K 100) the normalized absorbances for B and C reach limiting values whose ratio corresponds to that predicted by the equilibrium constant. Again as in the previous case, the Ac' = A x \ holds during the entire relationship An.\ course of the experiment and the kinetic parameters can be determined by following either B or C . Digital simulation was also used to check the two EC
+
2374
= A z S = (Azo(),)
-
cases. Since Z is in excess (for example Cz/C,\ > 1000), care must be taken not to let the value of kfCBCZor kbCc exceed 0.1 in the diffusion equations (Equations 30 and 31). Consequently, a much greater number of calculation loops must be used in order to simulate large values of It. Results indicate that the simulation technique agrees a t all points with the rigorous solution to within 1%. Case IV-Reversible Chemical Reaction Preceding Charge Transfer (CE). This case can be represented as:
A * n e + B (34b) The classical case in which A and B are electroactive and Z inactive at the applied potential has been solved in terms of the i-t relationship with the assumption K
O,X
t
>
2 O;C,/Cz
-----)
= K,C,4
";C,h/Cz
0,s = O;C,\
+ C,
= C*
e CZ*,CB = 0 (38a)
+
-
K,C,4 C, C* Y C,*,C, -+
-----t
=
O,C,
=
f(t),C,
=
0
(38b)
f ( t ) (38c)
Like the previous case, change of variables must be made by letting C = Cz + CA and U = CA - K Cz before attempting the solution. The solutions obtained are valid only when K < 10W3. A working curve based on Equations (18) J. Koutecky and R . B. Brdicka. Collect. Czech. Chem. Comrnun.. 12, 337 (1947). (19) P. Delahay. "New Instrumental Methods in Electrochemistry." Interscience. New York, N.Y., 1954.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 14, DECEMBER 1973
11 to 13 is shown in Figure 5. It can be seen that when k f t is small, not enough B is produced. The build up of B becomes diffusion limited and no longer depends on the prior equilibrium when k f K t > 103. The dynamic range of the working curve is 0.1 < h f K t < lo2. Since K 10.0
C
D E F G
(K =
kfCZ/kb;for Case V refer to Curve G only)
shown in Table I, Equation 15. From Equations 15 and 16, note that
and (42) where C* = Cz* + C,* is the total analytical concentration and I = k t hb, or Equation 42 can be rearranged to
+
From Equations 45 and 46, we see that lim A.A.\ = lim AB*\ = 1 hi)-0
k{-0
and The plot of AH.\' us. log Kzlt will have the identical shape as in Figure 5. From the above equation, one would expect to see that AB.\' /AB.\ approaches 0.5 a t large values of It if K = 1. This behavior can be easily rationalized by noting that at K = 1, the concentrations of Z and A are equal and thus CL* = C*. Case V-Charge Transfer Followed by Irreversible Catalytic Reaction. A f n e - B ki
Z + B - A + Z '
(44a)
(44b) where Z and Z' are in excess and electroinactive at the applied potential. The potentiostatic i-t relationship for the irreversible catalytic case has been obtained more than two decades ago (19). The potentiostatic A - t relationship was not available until recently. Kuwana and coworkers obtained a rigorous solution for the first-order case (8) and the second-order case was solved by digital simulation (6). For the first-order case, the analytical solution for the starting material A which has not been presented previously is
The plot of A4.\ and A B \ us. log h f ' t is shown in Figure 6 (curve G ) . Case VI-Charge Transfer Followed by Reversible Catalytic Reaction.
A f ne
-
B Elw
(4W
ki
B + Z - A + Z ' ki,
(4%)
This case implies the existence of the following reaction
Z f ne
Z' Elo'
(49)
From thermodynamics the equilibrium is governed by the relation:
lOg(k,/k,) = n(Elo' - EI0')/0.059 (50) If the equilibrium constant K for Equation 48b is large, then Reaction 48b proceeds spontaneously to the right. Therefore from Equation 50, it is implied that Z might also be electroactive at the applied potential corresponding to the reduction or oxidation of A. Since Z is in large excess, it is essential that the heterogeneous rate constant, k r . h r for
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 14, DECEMBER 1973
2375
10 from a totally irreversible catalytic reaction ( k , = 0 ) . The relationship, A A N = ABN, is again maintained during the entire course of electrolysis. One would expect, as shown, that as the CZ/C,\ ratio decreases (i.e., approaches a second-order condition), this above mentioned phenomenon will occur sooner for a constant value of K . Case VII-Charge Transfer Followed by SecondOrder Irreversible Disproportionation.
i$0,-
,c 06-
4
:‘I ,
A f n e - B
0s
(554
k
O-,o
-*i
-?C
L
LOG
lkll
Normalized absorbance-time relation for species B and C. Disproportionation reaction (Case V I I ) Figure 7.
A .v
Curves
A B
AB.’ AcS
=A ~ ( X , ~ ) / ~ ~ B ( X ) C , * ~ / ~ ) = A c ( X , t )/ ( 2 ~ c ( h ) C . 4\‘&T) *
2 B - A + C (5%) This sequence is second-order analogous to Case V. The potentiostatic i-t relationship has been solved previously with the aid of “reaction-layer’’ theory (20). Since no analytical solution is possible by conventional Laplace transformation technique for potentiostatic A-t relationship, the working curves (shown in Figure 7 ) , are obtained by digital simulation based on the following planar diffusion equations:
(56)
the Z/Z’ couple be negligible with respect to that of the A / B couple at the applied potential. Under these conditions, essentially a negligible fraction of the total current will be consumed on Reaction 49. With the above stated criteria and with the following planar diffusion equations and boundary conditions
and (541 One can easily establish the working curves (Figure 6) by means of digital simulation as analytical solutions are not available. The curves in Figure 6 are plotted by assuming species Z is in‘ 1000-fold excess of starting material A-ie., a virtually pseudo first-order condition. For any value of the equilibrium constant K , the system behaves like a simple charge transfer reaction ( i . e . , only diffusioncontrolled process exists) when hfC,t is smaller than 3 x 10-2. As the value of kfC,t increases, the effect of kinetics begins to show up. The competition between the diffusion and kinetic processes produces a minimum for A-\ at certain values of kfC,t depending upon the magnitude of K . As k f C L t further increases, the diffusion-controlled process once again takes over and the system is eventually indistinguishable from simple charge transfer. These are the characteristics of a catalytic reaction. One can easily rationalize this phenomenon by knowing that as the forward rate constant of the homogeneous chemical reaction increases to a very large value such that the regeneration of species A is virtually instantaneous and thus the reduction (or oxidation) of A to B becomes the ratedetermining step. However, this phenomenon will not be observed when the equilibrium constant becomes greater than -10. This is demonstrated in Curve G of Figure 6. In other words, one cannot tell the difference between the pseudo first-order reversible catalytic reaction with K >
-
2376
(57)
The useful dynamic range of the working curve is similar to the irreversible catalytic case (0.3 < ht < 10). It should be noted that A A Z = A B \ + Ac.’. The origin of the shape of the curve is obvious and will not be repeated. Some porphyrins have been shown by spectroelectrochemistry to follow this type of mechanism ( 2 2 ) . Case VIII-Disproportionation Reaction Followed by a n Irreversible Homogeneous Chemical Reaction.
A f n e - B k 2B A + C
(60a) (60b)
kb
k
C - D (60~) This is an extension of the previous simple disproportionation case. Since the intermediate chemical step (Equation 60b) is assumed to be reversible, one cannot exclude the possibility of a second charge transfer step. This is especially important as K for Reaction 60b approaches unity. B * n e - + C (61) The reader should refer to the original literature by Marcoux (22) on the discussion of the differentiation be-
(20) E. F. Orleman and D. M . H. Kern, J. Amer. Chem. Soc.. 7 5 , 3058 (1953). (21) B. P. Neri and G. S. Wilson, Anal. Chem., 45,442 (1973). (22) L. Marcoux, J , Phys. Chem.. 76, 3254 (1972).
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 14, DECEMBER 1973
I
1
I
I
,
I
O' I 06
P
..c
I
OP
~
08
~
;.
I
..I,
.#\
slo
k
.d,
.lo LOG
L t G
d,
,lo
j0
\I,
l',l
,,I
Figure 9. Normalized absorbance-time relation for species C. Figure 8. Normalized absorbance-time relation for species B. Disproportionation reaction with K = 100 and K = 1 followed by Disproportionation reaction followed by irreversible chemical irreversible chemical reaction (Case VI1 I ) . Ac-' = A c ( X , t ) / reaction (Case V I I I ) . AB.' = A B ( X , ~ ) / ( ~ C B ( X ) C A * ~ / ~ )
(2€C(X)C,4*rn/fi)
Curves A B
C D E F G
K 0.01 0.01 0.01 1 1 100 100
k
K 100 1 1
Curves
0
A B C
10 k f 100 k f
0 10 k f 0 10 k f
tween the ambiguity of an ECC and an ECE mechanism, and by Kuwana et al. (7) on the discussion of spectroelectrochemical properties of EEC mechanisms. Before attempting to consider the second electron transfer complication, it is more logical for us to present the solution under the assumption that Reaction 61 does not occur. The prerequisite for this assumption is similar to the discussion under Case VI. Since no readily available analytical solution exists, the potentiostatic A-t relationship is obtained by digital simulation based on:
k 0 0 10 k f
I OOIQC ..\+
'"" [:::o 00%
0 0 0 4 ~ 0002
-
I -10
-15
-2:
I -45
-80
-05
I
1
I
0
05
80
1 .I
1 10
d G
Figure 10. Normalized absorbance-time relation for species C. Disproportionation reaction with K = 0.01 followed by irreversible chemical reaction (Case VI I I ) . Ac" same as Figure 9 Curves
A
B C
k 0 10 k f 100 kr
and and C can be used to evaluate the rate parameters. In practice, unless species C has a molar absorptivity a t least 50 times greater than that of B, one cannot use Figure 10 for rate measurement when K i 0.01. In other words, when K i 0.01, Figure 8 appears to be more useful than Figure 10. For K 1 100, one would use either Figure 8 or Figure 9 with equal facility.
and
The normalized absorbance of B us. log It with varying equilibrium constant, K , and the rate constant, h, for the subsequent chemical reaction is plotted in Figure 8. The normalized absorbance of C us. log It with varying K and h is shown in Figures 9 and 10 (the difference in scales should be noted). From Figure 8, one can clearly see that when both K and h are small ( e . g . , K = 0.01, h < 10 hf for It < lo), the system will behave like a simple charge transfer reaction and when K becomes larger (e.g., K > 100, h 5 100 k f ) , the system will then resemble a simple disproportionation reaction. Theoretically both the normalized absorbance plots of B
DIAGNOSTIC CRITERIA It is by no means our prime objective to report an additional method for measuring the kinetic parameters of a reaction, rather to make use of additional information which may be available from the experiment. From this the mechanism of a light-absorbing electrochemical reaction coupled with a homogeneous chemical reaction can be assigned with reasonable confidence. Assignment of Mechanism. In every case, the absorbance of each individual species depends exponentially on the reaction rate constants. The most convenient method of displaying the experimental results has been shown to be the plot of normalized absorbance (8) of a species of in-
ANALYTICAL CHEMISTRY, VOL. 45, NO. 14, DECEMBER 1973 * 2377
Table II. Summary of Diagnostic Criteria, dAiV/d log (kt)a Reactant
Intermediate
Species
Case I Case I I Case I I I Case I V Case V Case VI Case V I I Case VI I I
A A A Z A A A A
0 0 0 4-0.31 -0.52 -0.52 ...
...
Product
-
Species
Species
... B B A
B B B B
Restrictions
...
B
0
-0.71 -0.71 4-0.31 -0.52 -0.52 -0.48 -0.48
C C B
4-0.71 $0.71 4-0.31
None None K > 100 K 10
C C
None See text
f0.40
4-0.40
a For the definition of normalized absorbance refer to the legend in the figure.
terest us. the logarithm of time (or actually the product of rate constant and time). By normalized absorbance is meant the ratio of the observed absorbance a t time t to the absorbance under pure diffusion-controlled conditions without chemical complication. This is similar to the familiar plot of kinetic current us. diffusion current in a conventional polarographic experiment. By comparing an irreversible EC mechanism with an irreversible catalytic process, one would intuitively realize that the rate of disappearance of the intermediate (or species B) is faster in the former case than in the latter. This is because in the catalytic case, B is replenished through a cyclic process and in the EC case B is consumed by a succeeding chemical reaction. One could thus in a semi-quantitative manner establish criteria for distinguishing these two processes. By measuring the slope of the working curve for the intermediate (this is equivalent to the measurement of the rate of disappearance of the intermediate), one finds the slope for the EC case to be -1.4 times larger than that of a catalytic case. (Finding the slope at the inflection point can be best achieved by Fortran programming). If the reaction is second-order in nature and its equilibrium constant is small, one could simply increase the substrate concentration until the system behaves like a pseudo first-order reaction and the apparent forward rate constant becomes much larger than the reverse rate constant. Naturally, one should justify this method by his particular experimental condition. Following the similar argument, one could foresee that the disappearance rate of the intermediate for a second-order disproportionation reaction is even smaller than that of a pseudo first-order irreversible catalytic case. However, the difference is about 8% and it would probably be difficult to distinguish these experimentally. In the CE case, because of the restrictions, direct comparison is not meaningful. However, note that the slope for the intermediate is positive and the slope for the intermediate of every other case is negative. The results of the measurements are summarized in Table 11. As mentioned before, it is not necessary to measure the absorbance of an intermediate. It may be easier to measure the absorbance of a starting material or of a final product. It goes without saying that the choice is made depending on the experimental conditions, particularly the magnitude of the molar absorptivity and the location of the absorption peak. Nonetheless, for most cases, the measurement of either the reactant, the product, or the intermediate should equally suit the purpose. By knowing the time dependence of all species, it is possible to more easily verify the mechanism involved and to obtain spectral properties of reactive intermediates. 2378
Measurement and Significance of Total Absorbance Change. One frequently encounters the situation in which the spectra of the starting material, the intermediate, and the final products overlap with each other so that it is difficult to choose a wavelength at which only one species absorbs. It is then necessary to consider the total contribution from all the absorbing species. In addition, one sometimes would like to calculate the spectrum of an intermediate in a multicomponent system provided that the mechanism and kinetics are known. Ordinarily the experimental results reflect the absorbance changes for all species and from this information spectral and kinetic information must be obtained. The total absorbance of an ( m n) component system ( m = number of starting species and n = number of intermediates and/or products) is:
+
AdA.t)
=
TAs(A,t) y=1
+
FA,IA,t)
(67)
1'
where S denotes starting species and j denotes intermediates and/or products. The total absorbance change at time t is:
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, N O . 14, DECEMBER 1973
rn
where AsO denotes initial absorbance ( t = 0). Based on Equation 68, the results for Cases I-V for which exact solutions exist are summarized in Table 111. For all cases, the normalized function of total absorbance change can be used to describe the significance of each individual equation. A generalized representation of all cases can be written as:
A set of working curves based on L4.l us. log It for various values of 0 can be constructed. Kinetic parameters can be determined by curve fitting if the log k t (theoretical) and log t (experimental) plots and all other constants are known. I t is interesting to note that = 1
lim AA,.” t - m
AAT.\‘ = AA,(X,t)/Ut-
D t
(74)
fi
where St = cf(X) - ts(X)--i.e., the difference between the molar absorptivity of the final products and that of initial starting species. [ t f ( X ) and ts(X) are sometimes weighted average values of molar absorptivity. See Case IV.] Note that no knowledge of c for the intermediate is required in establishing Equation 74. Case I. Equation 69 can be rearranged to obtain
- t*(X))-
m
+ 1 +1 KO ~
Thus, a t sufficiently long times, the steady state absorbance value is independent of kinetics and the equilibrium constant can be evaluated directly from Equation 80. The behavior of species Z is similar to Case I1 and will not be repeated here. Case IV. The normalized total absorbance is:
AA~.’
O t
-
= AA~(A,~)/~c~*-[~B(X)
fi
(tz(h)
+
(75)
KtA(X))I= ; f i d ( k f , K ,t ) (81)
A plot of l A T . v us. log t is unambigously a straight line with a slope equal to zero (note D = DA).However, this equation will obviously become unusable if ciz = t g (an isosbestic point). Case 11. The total absorbance change is given by:
I t should be noted that there are two initial species Z and A present in the solution. t f in Equation 74 becomes the weighted sum of t L ( X ) and cA(X) in Equation 81. Since K is much smaller than one, unless e Z is exceedingly small, the absorbance contribution from A is often negligible with respect to Z. The shape of 1A’ us log k f K t working curve is identical to Figure 5 . Kinetic parameters can then be evaluated with the aid of this plot. Obviously the KcA(X).The steady state restriction is m ( X ) # cz(X) value of the total absorbance is:
AATS = AAT(X,t)/2(tg(X)
AA,(h,t) = A,(X,t)
+
AB(X,t)
Az(X,t) -
fi
= 1
+ Ac(A,t) +
(AAO(X)
+
AzO(X)) (76)
and
A,(X,t)
= AZYX)
+
(77)
Equation 77 arises from the fact that Z is in large excess and electroinactive. Therefore, there is no substantial change in the concentration of Z. The normalized function is:
(tz(h)
+
K~A(X))I = 1 (82)
Thus, like the previous case, it is possible to determine the equilibrium constant without the knowledge of kinetic informat ion. Case V. Comparing Equations 45 and 46 with 73, the relationship below is obtained. A A A = ABS = AAT.V =
Since limt,, -y(kf’,t) = 0, it follows that limt, AAT~ = 1. That is, a t sufficiently long times the value of AAN is independent of the rate of the reaction and the system behaves like a simple charge transfer reaction. A plot of UT‘us log k,’t (or log t ) should yield a curve which asymptotically approaches one from an initial value which can be either positive or negative. This results because the value of 6’ can be either positive or negative depending on the magnitude of t A ( X ) , tg(X), and cc(X). Kinetic information can be obtained by constructing a set of working curves for various values of 8. However, kinetic information is lost if tg(X) = t c ( X ) and/or c c ( X ) = t A ( X ) . If k f ’ , tA(h), and c C ( X ) are known, one can back calculate tB(X) for the intermediate with the aid of Equation 78. One can see that the difficulty in measuring a small absorbance change on top of a large initial value may exist if the last two terms in Equation 76 are considerably larger than the first three terms. Situations like this should be avoided. Case 111. If Equation 71 is rearranged:
As Kuwana has shown ( 8 ) , kinetic information can be evaluated from the steady state total absorbance value:
If Z and Z‘ absorb, then the absorbance change is given by: AA,’(h,t)
+
= AAT(h,t)
Az’(X,t)
(85)
Note that Azo(X) - AZ(X,t) N= 0, because Z is in excess. Az’(X,t) in Equation 85 can be, in principle, calculated from:
Az’(X,t) = t~’(h)iZ~S‘SYCli(X,t)drdf (866) 0
0
For cases in which digital simulation was used, the construction of normalized total absorbance working curves and treatment of data are quite straightforward. For any electron transfer process followed by homogeneous chemical reaction(s), the normalized absorbance of an initial starting species, ( v i d e supra), is expressed as:
ANALYTICAL CHEMISTRY, VOL. 45, NO. 14, DECEMBER 1973
2379
AATN = f ( k , t )
and the normalized absorbance of any species j that is the product of first-electron transfer reaction or is involved in subsequent chemical reaction is defined as:
(89)
For every value of k t , there is a corresponding value of rlAT.''. After the working curves (&IT" us. log k t ) are established, it becomes a simple matter to determine kinetic or nonkinetic parameters. Obviously, J ( k , t ) can be expressed in closed form if exact solutions exist.
ACKNOWLEDGMENT We would like to thank S. W. Feldberg for his many helpful discussions and advice. To T. Kuwana and his associates, we are indebted for programs as yet unpublished. From simulation working curves, i.e., A s N and Ajlv us. log k t and Equations 87 and 88, one can calculate AsO(X) - As(X,t) and A,(X,t) a t time t. With the aid of Equations 67 and 68, one can obtain a function f ( k , t ) in the form of Equation 74 and its corresponding numerical values.
Received for review March 19, 1973. Accepted July 12, 1973. This investigation was supported in part by National Science Foundation Grant GP-28051 and the Office of Naval Research.
Selected Trace Metal Determination of Spot Tape Samples by Anodic Stripping Voltammetry Kathryn E. M a c L e o d and Robert E. Lee, Jr. Quality Assurance and Environmental Monitoring Laboratory, U.S. Environmental Protection Agency, National Environmental Research Center, Research Triangle Park, N.C. 2771 7
As part of a program to determine the contribution of trace metals in fuels to ambient air levels, anodic stripping voltammetric analysis was applied to two-hour samples collected with an AIS1 spot tape sampler. The sampled spots were analyzed for lead, cadmium, and copper after ashing at low temperature and extracting with acid. The quantity of metal present in the samples ranged from 7 to 350 ng of cadmium, from 80 ng to 2.4 pg of lead, and from 6 ng to 1 pg of copper. The maximum relative standard deviation for the method was less than 12% for all three metals with the averages of 4.5% for Cd, 5.9% for lead, and 3.7% for copper. Analytical sensitivities were sufficient to characterize diurnal variations of these metals in samples collected in Chicago and Washington, D.C.
Anodic stripping voltammetry is used routinely in our laboratory for the determination of trace concentrations (ppb level) of metals in fuels, biological samples, and other environmental materials. As part of a program to determine trace metals in motor vehicle fuels, we examined the feasibility of applying ASV analysis to ambient air samples collected adjacent to highways where suspended particulate matter is composed primarily of fuel combustion products. Samples were collected with AISI tape samplers, which are commonly used in many air pollution programs to determine the soiling index (coefficient of haze) of suspended particulate matter. The analytical procedures described in this report permit the determination of the soiling index and of ambient concentrations of lead, cadmium, and copper in two-hour samples. This method can easily be extended to measure other ASV-responsive metals including Zn, Bi, Ag, T1, and Sb.
The determination of particulate metals suspended in air is usually limited to a 24-hour sampling period using a hi-volume air sampler ( I ) . Continuous methods for trace metal determinations are currently in the research stage and are not available for routine measurements. A major problem in successfully reducing the sampling time to derive information on diurnal trace metal patterns has been the insensitivity of available analytical methods. Among several trace metal analytical techniques that have recently become available, anodic stripping voltammetry (ASV) offers the required sensitivity and the capability for multielement analysis (2).
EXPERIMENTAL
Robson and K. E. Foster, Amer. Ind. Hyg. Ass., J., 24, 404 (1962). (2) W. R. Matson, R . M . Griffin. and G . B. Schreiber, "Trace Substances in Environmental Health-IV," University of Missouri, 1971, p 396.
Apparatus. Samples were collected with AISI tape samplers (3) located a t EPA's Continuous Air Monitoring Project stations ( 4 ) in Chicago and Washington, D.C. The AISI tape sampler determines the soiling property of suspended particulate matter by means of a n absorbance measurement. Ambient air is drawn through a circular portion 1 inch in diameter of a continuous strip of Whatman No. 4 filter paper 2 inches wide. T h e sampling interval used in this study was 2 hours, after which a n automatic mechanism advanced t h e tape t o expose a clean portion of t h e filter. The optical density of the sampled portion of the tape is measured by reference t o a clean unsampled portion of the tape used a s a zero reference. Results are generally reported in units of coefficient of haze (Coh), which is defined as t h e quantity of
(1) C. D.
2380
(3) ASTM, "ASTM Standards on Methods of Atmospheric Sampling and Analysis," 1962, p 497. (4) G. A . Jutze and E. C. Tabor, J. Air Poliut. Confr. Ass., 13, 278
(1963).
ANALYTICAL CHEMISTRY, VOL. 45, NO. 14, DECEMBER 1973
