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 pnRin 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).
The Journal of Physical Chemistry, Vol. 86, No. 10, 1982
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4
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Tuttie et ai.
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Flgur 1. comperlsons between solvated electron absorption spectra in llquld ammonia obtalned from metal solutions with those obtained from pulse radiolysis under nearly the same conditions. Reduced absorbance, F ,is plotted vs. frequency, Y*, on the r e f e r " frequency scale (seeref 2 for relevant detalk). Root-mean-squareddevktbns, AF,are plotted below each graph. A: (0)data of Rublnsteln at 198 K; (A)data of Jou and Freeman at 200 K shifted by -33 Cm-'. B: (0)data of Rubinstehr at 218 K (A)data of Jou and Freeman at 220 K shifted by -32 cm-I.
their data, Jou and Freeman concluded that small but significant changes occured in the shapes of their spectra in ND3and probably also in NH3 as temperature increased. This contrasts with the constancy of shape, referred to as shape stability, recently demonstrated2for solvated electron optical absorption bands on changing temperature and/or pressure in a number of polar solvents, including NH3 and ND3. Some deviations3 from the aforementioned spectral shape stability appear to be shown by solvated electron spectra in H 2 0 and D20. In each of these liquids the spectra are, nevertheless, accurately representable as linear combinations of only two linearly independent functions each of which has a temperature-independent shape.4 As a result, these spectra provide examples of an extended shape stabilitp of their component bands. It remains to be seen whether or not the small deviations from spectral shape stability noted in ND3 and possibility NH3 by Jou and Freeman are legitimately to be accounted for in such terms. To examine this point in the case of ammonia, we have combinated the data for Rubinsteid with those of Jou and (2)T.R. Tuttle, Jr., and S. Golden, J . Chem. SOC.,Faraday Trans. 2,77,873 (1981). (3) F.-Y. Jou and G. R. Freeman, J. Phys. Chem., 83, 2383 (1979). (4)T. R. Tuttle. Jr.. and S. Golden. J. Phvs. Chem.. 84.2457 (1980). (5)5.Golden &d T.R. Tuttle, Jr., 2. ChimeSOC.,Furaday Trans. 2, 77,889 (1981).
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Flgwe 2. Comparisons between solvated electron absorption spectra in llquld perdeuterbammonla obtalned from metal sdutbns with those obtained from pulse radiolysls under nearly the same conditions. Reduced absorbance, F ,is plotted vs. frequency, v*, on the reference frequency scale (see ref 2 for relevant details). Root-mean-squared deviations, AF,are plotted below each graph. A: (0)data of Hurley from sodlum solutions at 202 K; (A)data of Hurley from potassium solutions at 202 K shifted by 22 cm-'; (V)data of Jou and Freeman at 200 K shifted by -55 cm-I. B: (0)data of Hurley from sodium solutions at 217 K; (A)data of Hurley from potassium soiutlons at 217 K shifted b 42 cm-I; (V)data of Jou and Freeman at 220 K shifted by -7 cm- . C (0)data of Hurley from sodlum solutions at 239 K; (A)data of Hurley from potassium solutions at 239 K shifted by 37 cm-'; (V)data of Jou and Freeman at 240 K shlfted by -45 cm-'.
Y
Freeman. In the case of perdeuterioammonia, the data of Hurley' are combined with those of Jou and Feeman. The additional data permit comparisons between solvated electron spectra obtained independently at or near the same temperature and pressure. Two such comparisons (6) G. Rubinstein, Ph.D. Thesis, Brandeis University, Waltham, MA, 1973. (7) I. Hurley, S. Golden, and T. R. Tuttle, Jr., "Metal-Ammonia Solutions", Butterworths, London, 1970,p 503.
The Journal of Physlcal Chemistry, Vol. 86, No. IO, 1982 1803
Solvated Electron Optical Absorption Bands
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F~QWO 3. Average spectral proflle for sobated electron spectra In llqukl ammonla between 198 and 255 K. Reduced absorbance, F ,Is plotted vs. frequency, Y*, on the reference frequency scale (see ref 2 for relevant details). Root-mean-squared devlatkns, AF,are glven below the spectral proflle. The values of the shlfts, A, are as follows. Rublnstein's data: at 198 K, A = 0 cm-'; at 208 K, A = 197 cm-'; at 218 K, A = 407 cm-'. Jou and Freeman's data: at 200 K, A = -33 cm-'; at 220 K, A = 363 cm-'; at 240 K, A = 784 cm-'; at 255 K, A = 1051 cm-'.
for ammonia are given in Figure 1 and three for perdeuterioammonia in Figure 2. In each comparison the spectra were shifted to minimize the root-mean-squared differences of frequencies at fixed relative absorbances using a procedure previously describedS2This procedure allows comparison of the profiles of the absorption bands by compensating for the relative shifta of the spectra which may occur without change in shape. All of the comparisons in Figures 1and 2 show detailed good agreement between independently determined solvated electron absorption spectra. In particular, the spectra obtained by pulse radiolysis' and those obtained by extrapolating spectra of alkali metal solutions to infinite dilution6*'are in excellent agreement with one another. For the most part the root-meanquared deviations, shown at the bottom of each graph, do not exceed 0.01 reduced absorbance unit. However, the deviations approach and even slightly exceed 0.02 reduced absorbance unit on the low-frequency side of the band. This is just the region where Jou and Freeman find the small changes of spectral shape. The selected spectral data for solvated electron in NH3 at various temperatures were shifted, as described: and their average was determined. This average spectrum is shown in Figure 3, together with the root-mean-squared deviations in relative absorbance. Clearly, throughout the spectral range down to about 6000 cm-', the deviations obtained for the shifted spectra at different temperatures do not exceed those shown by the different spectra at the same temperatures in Figure 1. At lower frequencies, the deviations shown in Figure 3 do exceed those shown in Figure 1 by as much as a factor of 2. This lends support to the small changes in shape noted by Jou and Freeman. However, because most of the points lower than about 6500 cm-' have been obtained with the aid of extrapolations of some of the experimental data at higher frequencies, these small changes in shape may arise largely as an artifact of the extrapolation procedure. Analogous results for the data selected in ND, at different temperatures are shown in Figure 4. (The complete data set is provided as supplementary material. See
A F 002
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Flgure 4. Average spectral profiles for solvated electron spectra in liqukl perdeuterioammonla between 200 and 246 K. Reduced absorbance, F,Is plotted vs. frequency Y O , on the reference frequency scale (see ref 2 for relevant details). Root-mean-squared deviation, AF,are glven below the spectral profile. The values of the shifts, A, are as follows. Hurley's data: at 202 K for Na, A = 0 cm-', for K, A = 22 cm-'; at 210 K for Na, A = 161 cm-', for K, A = 195 cm-I; at 217 K for Na, A = 333 cm-', for K, A = 375 cm-'; at 225 K for Na, A = 491 cm-', for K, A = 534 cm-'; at 232 K for Na, A = 661 cm-', for K, A = 706 cm-'; at 239 K for Na, A = 817 cm-', for K, A = 855 cm-I; at 246 K for Na, A = 947 cm-', for K, A = 974 cm-'. Jou and Freeman's data: at 200 K, A = -55 cm-'; at 220 K, A = 326 cm-'; at 240 K, A = 772 cm-'.
paragraph at end of text regarding supplementary material.) Here the deviations are very similar to those shown by different spectra at the same temperature (see Figure 2). Accordingly, any changes in the shape of solvated electron spectra which may occur in ND3 at different temperatures are comparable to or smaller than the experimental uncertainties. Within the experimental uncertainties, currently available, selected experimental data thus show that solvated electron absorption spectra display shape stability in ND3 at temperatures ranging from 200 to 246 K and probably also in NH3 at temperatures ranging from 198 to 255 K (under orthobaric conditions). The variations in spectral shape which must accompany the different temperature coefficients of the different parts of the band reported by Jou and Freeman appear to be within experimental error for the extended data used above. Much more accurate and precise spectral data than are now available over a wider range of temperatures are required to resolve the question of whether small changes in shape which correlate with changes in temperature actually do occur or not. For the present, the claim by Jou and Freeman that different kinds of transitions account for the absorptions on the two sides of the bands must be viewed as rather speculative. In fact, because the observed spectra in both ND3 and NH3 appear to display shape stability, most, if not all, of the solvated electron absorption bands in these two solvents would seem to result from bound to continuum transitions5with little, if any, contribution from bound to bound transitions. Supplementary Material Available: Tabulated absorbance values for the sodium and potassium solutions in ND3;these data were also used previously in establishing spectral shape stability in ND3 solutions2 (15 pages). Ordering information is given on any current masthead page.
