J. Phys. Chem. 1982, 86, 1794-1801
1794
recorded intensities of fast and slow components, respectively, extrapolated to zero time. The rate parameters are indicated in Figure 5. a represents the fraction of molecules which from t-S, isomerize and reach p-S,. If p-So is nonabsorbing, j3 equals unity. Using eq 1 and 2 in the case of nonabsorbing p-So, one can extract the individual rate constants, k, and kic, if a is known. There also is the possibility that kic ElI2+ 0.2/n (V) and E2 < El - 0.2/n (V). Actual values were as follows: (Fe(II1)) = 0 V, Ez = 4 . 5 V; (Pb(II)) El = 4 . 3 V, E2= -0.7 V. The SMDE drop was allowed to grow and stabilize for a minimum of 1.5 s at the delay potential E,. For recording of iNpand iRp,the gain of the current to voltage (i-E) converter was optimized to provide maximum resolution for the 12-bit analog to digital conversion (i.e., such that no currents were less than 25% of full scale output voltage (A10 V)). Data points were sampled and converted every 1.0 ms in the range t = 1.0-100.0 ms (for i~p) and in the range t - tl = 1.0-100.0 ms (for iw and iE). In all cases the gain optimization caused the first one to five current points (at 1-5 ms) to fall out of the range of the analog to digital converter and to voltage limit the i-E converter. (Loss of potential control and amplifier saturation were prevented by a diode protection network in the feedback loop of the i-E converter). Data points from 1to 5 ms were thus ignored in the analysis procedure. For recording of the dc current the gain of the i-E converter was reoptimized, since iE was typically much smaller than either iNp or iRp,especially for long values of tl. Instantaneous current sampling was extremely sensitive to spurious pickup of power-he transients and 60-Hz noise, even at the high analyte concentrations employed. The use of signal averaging was an absolute necessity, and all data recorded in this study represent the average of 20 double pulse experiments. Depletion and other history effects were eliminated by dispensing and dislodging a series of drops at the delay potential (E,) prior to each double potential step experiment. The effectiveness of this procedure was evaluated at large values by examination of the reverse currents (iw) of tl (5 s). Noticeable (but irreproducible) “first drop effects” (on the order of 1-5% of iw at t - tl = 50 ms) were noted for both Fe(II1) and Pb(I1) if this procedure was not followed. If three of more drops were dispensed and dislodged between experiments, no significant drop-to-drop variations were apparent, and this procedure was employed for all experiments described here.
hl
Results and Discussion As a basis for analysis of spherical effects, we consider the planar, stationary electrode response for the general case of 0 + nee- pR R
-
-
0’ + nRe-
i.e., where 0 may be reduced to p moles of R, with no electrons, and R may be oxidized losing nR electrons to a new product 0’. If E2 and E , (see Figure 2) are chosen to be well into the diffusion-limited region for each reaction (thus driving the surface concentrations of 0 and R to zero at E2 and El, respectively), the chronoamperometric responses are given by2 iNp -1Rp
=
= n$ACob(Do/d)l/z
( t 5 tl)
(1)
, (2)
where iNpis the diffusion-limited NP current,
iRp
is the
Brumleve and Osteryoung
diffusion-limited RP current, tl is the time at which the potential is switched from E2 to El, F is the value of the Faraday constant, Cob is the bulk concentration of reactant, 0, A is the electrode area, and Do is the diffusion coefficient of 0. The diffusion-limited dc current, iDc,is also given by eq 1, the only distinction being that t > t,. A plot of -iRP/iwvs. [ t / ( t- tl)]’/* - 1 is thus linear with a slope (S-RpIDc) of pnR/no.2 The particular case of pnR/no = 1 applies to a wide variety of reactions.2 It is the case for simple reversible reactions in which product 0’is identical with reactant 0, no = nR = n, and p = 1. In this case a plot of iDc- iRp vs. l / ( t - t , ) ’ / 2 is linear with a slope (SDC-Rp) of nFACob(Do/7r)1/2.The ratio of this slope to the slope (S,) of a plot of iNp vs. t-ll2 is thus unity for diffusion to a planar, stationary electrode. This may also be expressed as2 iNP(t
- tl) = iDc(t) - iRP(t)
(t > tl)
(3)
When the predictions of the planar diffusion theory are applied to data obtained at spherical electrodes, however, significant deviations may result. These effects have been noted in the literature for the DME2 and are illustrated here for a stationary spherical electrode in Figure 1. Here tl = 1s and the pulse current is sampled at the effective value of 41.7 ms, so t = 1.0417 s. For the case of Fe(III), the measured current iRpagrees with that calculated from iE, iNp,and eq 3 within 0.3%. But for Pb(II), the value of iRpexceeds that predicted by 19%. In the case of Fe(III), the product, Fe(II), is soluble in solution, and deviations from eq 3 due to spherical diffusion tend to cancel. In the case of Pb(II), the product, Pb(Hg), diffuses into the electrode, and spherical diffusion effects tend to be enhanced. The difference in wave shape for the two systems is due to the different number of electrons involved (n = 1 for Fe(1II) and n = 2 for Pb(I1)). Note the coincidence of half-wave potentials in the NP and RP scans in both cases as expected for reversible couples.lS2 Since one is concerned primarily with the effect of sphericity on the limiting currents iRp, iNP,and iDc (and not with small deviations in wave shape and position), the problem is best approached with double potential step chronoamperometry as shown in Figure 2. The choice of potentials El and E2is based upon the NP and RP polarographic data of Figure l. Figure 2 shows the raw, unsmoothed current-time responses for Pb(I1) for various values of tl as noted. The spherical diffusion enhancement of iw - iRpis quite clear, especially at large values of t,. Even at tl = 0.1 s, the experimental value of -iw exceeds that predicted by eq 6 (at t - tl = 100 ms) by 25 % . Data for Fe(II1) are qualitatively similar, except that the quantity iDc- iRpdeviates from the planar theory to a smaller extent. The major factor inherent in the experimental design which permits a straightforward quantitative interpretation of spherical diffusion effects is the SMDE. Each Hg drop is allowed to grow and stabilize at zero Faradaic current at a delay potential (equal to El in this case) before the potential E2is applied. This assures that the potential E2 is applied at constant electrode area without the hydrodynamic complications encountered with similar experiments at the DME. This is particularly important for products soluble in mercury because internal stirring due to drop growth is eliminated during time t,. The reproducible drop area of the device makes it possible to average signals and therefore to achieve good signal-to-noiseratios. Simplification of the boundary conditions in this fashion permits the derivation of a general expression for the
The Journal of Physical Chemistty, Vol. 86, No. 10, 1982 1797
Spherical Diffusion in Reverse Pulse Voltammetry
20 r
TABLE I: Slope and Slope Ratio for Iron(II1) Oxalate
I
SNP,
SDC-RP,
r A s,'~ AS"^ Rexptl Rtheor %or 0.9000 0.9111 1.012 0.997 1.002 0.9000 0.9113 1.013 0.996 0.998 0.9000 0.9163 1.018 0.994 0.997 0.992 0.990 0.9000 0.9179 1.020 0.9000 0.9319 1.035 0.989 0.987 0.5 an alternative solution can be employed in which q1 is replaced by ql': Qi' = 1/[5Ui(?rti)1/2] 7 6 6 ~ 2 2 t 1 ' / ~ / ( ~ i ~ '3az2t1 /~) (9) The dependence of q1 and ql' on (DRtl)i/2/rois shown in Figure 3. There is a considerable range of (DFtl)1/2/ro around the value 0.5 where q1 and q< are equal in value, and therefore the use of eq 8 in eq 5 for (DRtl)'f2/roI0.5 and of eq 9 in eq 5 for (DRtl)i/2/ro> 0.5 provldes a solution valid over the entire range of t - ti < ti. As shown in Appendix 1,these equations are easily generalized to the
+
1.09 1.17 1.25 1.38 1.65 2.03
ro) or equal (r I ro) fluxes at the interface
f
(stl)1/2/[&(&
= DRCR(rO,tl)/(DOCOb)
m
cR(r,s) = (A"/r) sinh [ r ( ~ / D ~ ) l / ~ r] I ro
I/(nFADo) = U(s)DR(&/a2
a2)1
Expressing the term l/(v's T a2)in the sum as a series (valid for large v's)
1
= (A'/r) e ~ p [ - r ( s / D ~ ) l / ~ ]r
=F
The terms L1, L2, and L3 are easily inverted and can be found in tables of Laplace transforms. The inverse Laplace transform of the last term (L4) is representable only as a series solution (or as an integral equation, using the convolution theorem). Invoking the series approach, the asymptotic expansion for large values of (st1)ll2(see eq 7.1.23 of ref 12) gives
( ~ 6 f 11)
where 7
= q1 exp(-stl)/[az(&
L~ = B erf
(A71 B =u~/uI
(-416)
a2)l
L, = a2[exp(-stl) - 11/[s(v5T a2)l
For large values of d s / a z ,coth ( d s / a 2 ) 1,for which eq A5 may be inverted to give (for t I tl) (CObDO/DR)[(a2/al
F
(-1p
n=O
[(t - tl)/tl]n+1/21'(n+ Yz) X
5 [ i a 2 ( t- tl)1/2]pn/{I'[(n+ r + 3)/2]))
r=n
The equations for iNp and iDc (eq A23 and A24) are the well-known results for single step chronoamperometry in the case of spherical diffusion. We now consider the characteristics of eq A25, which gives the current i ~ p .For cases in which t - tl < tl/lOO, the term G ( t ) may be neglected. For diffusion both away from and into the electrode, the series expansions of eq A21 and A22 are valid only for large values of the argumenta involving s. It can be shown that the double series in G ( t ) is thus valid only for small values of t such that t - tl < tl. This is exactly the desired range of validity for the typical reverse pulse experiment in which the pulse widths (t - tl) are much shorter than time during which product is generated (tl). Equation A25 is thus valid for diffusion into solution as long as t - tl C tl. Additional restrictions apply when product diffuses into the electrode. These arise because of the simplifications made in deriving eq A6 and A13 (i.e., allowing coth v'sla,) 1). It can thus be shown that eq A25 applies only when
-
(12) M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions", National Bureau of Standards Applied Mathematics Series, No. 55, 1964, pp 295-329.
1801
J. Phys. Chem. 1982, 86, 1801-1803
(DRtl)lI2/roC 0.5 for r I r p It is possible, however, to derive an approximate expression for large values of (DRtl)1/2/ro in this case. By employing the first three terms of the asymptotic expansion for coth ( x ) , we can show that for small x l / [ x coth ( x ) - 11 3/n2 + 3/15 + ... (A26)
-
If q A 2 6 is substituted into eq A5, the resulting expression for CR(r0,s)may be inverted to yield a new function ql':
+ y5 + 6 ~ 2 ~ t 1 ' / ~ / ( ~ 1 &+) 3U22tl
41' = 1/[5Ul(7ft1)1/2]
(A27)
Figure 3 shows the behavior of q1 and q{ as a function of (DRtl)1/2/ro. Either equation is applicable near (DRtl)'l2/ro= 0.5. Thus, for diffusion into the electrode, eq A25 applies for (DRtl)1/2/roC 0.5 and applies for (DRtl)1/2/ro> 0.5 when q1 is replaced by q:. For the typical case t - tl C tl/lOO, it appears that eq A25 holds for (DRtl)1/2/ro > 0.5 with ql' used in place of q1 and F(t) and G ( t )neglected. This has been verified by comparison with the results of digital simulation but will not be discussed further because the experimental results in this paper are all in the region (Dtl)1/2/roC 0.5. For typical sizes of mercury drops, this will generally be the case. If DR = 6 X lo4 cm2s-l and ro = 0.025 cm, (DRtl)1/2/ro C 0.5 for t l C 25 s. Although the above treatment has been derived in terms of a reversible electrode reaction, the only requirements are that the surface concentrations of 0 and R be driven to zero a t potentials E2 and El, respectively. Equations A23-A25 may thus be extended to the general case of 0 + nee- pR and R 0' + nRe- by replacing n with no in eq A23 and A24 and by replacing n with pnR in eq A25.
-
-
Appendix 2 The complicated dependence of the chronoamperometric current on the value of DR requires a careful analysis of the dependence of errors in the derived value of DR on errors in the experimentally determined quantity R (eq 11). Here we analyze the error in DR arising from applying the procedure in the text for the case of diffusion away from the electrode when Do N Dw The result is then extended to the general case over a limited range of parameters.
When Do N DR, the terms F ( t ) and G ( t ) of eq A25 become negligible, and R = SDC-RP/SNP is given by
R = q1(Do/DR)1/2 The quantity to be determined, DRIDo, is thus given by
D d D O = q12/R2 (A29 Note that q1 is also a function of DR and Do. The relative error in DR/Do can be expressed as a(DR/Do)/(DR/Do) = 2%1/q1- 2 ( a R / R ) (A30) But for Do N DR aql/ql = [~(DR/DO)/(DR/DO)IH((DR~~)'/~/~ (A31)
where H ( x ) = exp(x2)erfc ( x ) . Substituting eq A31 into eq A30 yields
a(DR/Do)/(DR/Do) = W / [ 1 - H ( ( D ~ t i ) " ~ / r o ) l ) ( a R /(A32) R)
Thus the standard deviation of DR/Do, s h I D 0is , given by SD,/D,/(DR/DO)= {2/[1 - H ( ( D R ~ ~ ) ' / ~ / ~ ~ (A33) )]~SR/R where SR is the standard deviation of R . From the properties of H ( s ) (see eq 7.1.13 and Table 7.9 of ref 12), it is easily shown that for x C 0.5 1/[1 - H ( x ) ] C 1 . 3 / ~ (A34) and a more useful expression results for (DRtl)1/2/ro C 0.5:
S&/DJ(DR/DO)< [ & / @ ~ t l ) ' / ~ ] s ~ / R(A35) where K = 2.6. This result can be extended to the general case of diffusion of product either into or away from the electrode over the range 0.25 C DR/Do C 4 by changing the constant K to 3. This result was obtained by exhaustive calculation employing values of R into which errors were introduced. Since the final calculation of DR involves multiplying the ratio DR/Do by an independently determined value of Do, the final relative standard deviation in DR is given by
sD;/DR2 C (9r02/DRtl)sR2/R2 + sD;/Do2
(A36)
Thermal Effects on Solvated Electron Optical Absorption Bands in Liquids Ammonia and Perdeuterloammonia 1. R. luttle, Jr.," Sldney Golden, and Ian Hurleyt DepaItnWnt of Chemistry, Brandels Unlverslty. Weltham, Massachusetts 02254 (Received September 9, 1981)
The shapes of optical absorption spectra of solvated electrons in liquids ammonia and perdeuterioammonia are shown to be the same within experimentalerror under nearly the same conditions, in the same solvent whether these spectra are obtained from spectra of metal solutions or from pulse radiolysis experiments. In addition, solvated electron absorption bands in each solvent at different temperatures display shape stability. As a result, most, if not all, of these absorption bands would appear to result from bound to continuum transitions. Recently, Jou and Freeman1 have determined optical absorption spectra of solvated electrons in liquid ammonia in the temperature range from 200 to 255 K and in liquid 'State University of New York at Albany,Department of Biology, Albany, N Y 12222. 0022-3654/82/2086-1801$01.25/0
perdeuterioammonia in the temperature range from 200 to 240 K, with results similar to those of previous investigations (see Figure 2 of ref 1). However, by analysis of (1) F.-Y. Jou and G. R. Freeman,
0 1982 American Chemical Society
J. Phys. Chem., 85, 629 (1981).
