3804
J. Phys. Chem. 1984,88, 3804-3810
result of the fitting procedure is displayed in Figure 10. Although the fits with the linear combination is certainly much better than with the solvated electron spectrum alone, differences remain which appreciably exceed experimental error. Similar difference between best fits and experimental spectra are shown also for the other solutions of the initial stage of decomposition as can be seen from the rms deviations given in Table 111. Furthermore, the assumption of an initial presence of amide implies an inexplicable loss of total dissolved metal during the initial stage of decomposition as can be seen from the values of [MITgiven in Table 111. Also, the initial presence of amide in itself does not account for the apparent initial slower growth of the 322-nm band. It appears that an adequate accounting of the spectral changes during this initial period requires a more complete description of the decomposing solutions than has been employed up to now. Unfortunately, the paucity of available pertinent data make the results of further detailed analysis highly questionable. Nevertheless, examination of the spectral behavior during the initial stage of decomposition in terms of the redox model of alkali metal ammonia solutions suggests a possible explanation for the seemingly slow increase in amide absorption during this period. By using available values of the relevant equilibrium constants3 for the redox model for sodium solutions at -50 O C and the ion-pairing constant for sodium amide,1° it is estimated that the more concentrated solutions studied here contain appreciable (10) C. A. Kraus and W. C. Bray, J. Am. Chem. SOC.,35, 1315 (1913). (1 1) G. Rubinstein, T. R. Tuttle, Jr., and S. Golden, J . Phys. Chem., 77, 2872 (1973). (12) T. R. Tuttle, Jr., and S. Golden, J . Chem. SOC.,Faraday Trans. 2, 75, 1146 (1979). (13) M.Ottolenghi and H. Linschitz, Adu. Chem. Ser., No.50, 149 (1965). (14) R. R. Cuthrell and J. J. Lagowski, J. Phys. Chem., 71, 1298 (1967). (15) J. Corset and G. Lepoutre, J . Chim. Phys., 63,659 (1966).
concentrations of Na- containing species as well as solvated electrons. For example, in the most concentrated sodium solution about 30% of the sodium is dissolved as Na--containing species. This figure drops rapidly to less than 10% for the first solution in the intermediate stage of decomposition. These Na--containing species are known to absorb very similarly to solvated electrons in the infrared and visible. However, their ultraviolet absorption has not been observed previously. Accordingly, a possible explanation of the slow increase in amide absorption initially is that the Na- species has an absorption in this region whose initial rapid decrease masks the simultaneous increase in the amide absorption. After the initial period the Na- concentration becomes sufficiently small so that its absorption no longer has a substantial effect. In addition, the similarity of the Na- and S- absorptions in the visible can account for the success of the original assumption that the absorption of the metal solution was entirely attributable to S-. In this way the disappearance of metal encountered in attributing the initial absorption near 322 nm to amide itself is also avoided. We have determined solvated electron and sodium amide spectra in the visible and ultraviolet regions. The solvated electron spectrum passes through a minimum near zero at 325 nm and shows a modestly sharp rise in absorption near the solvent ultraviolet cutoff, although not so sharp as previously reported. An anomalously slow increase in amide absorption is attributed to the presence of an Na- absorption band. Although this last attribution is rather speculative it would appear to be a suitable subject for future investigation. Registry No. Sodium, 7440-23-5; ammonia, 7664-41-7.
Supplementary Material Available: The measured absorbances in the wavelength range 600-230 nm at 2.5-nm intervals for a decomposing sodium-ammonia solution (10 pages). Ordering information is available on any current masthead page.
Optical Absorptlon Spectra of Solvated Electrons In Mixtures of Ammonla and Methylamlne Catherine M. Stupak, T. R. Tuttle, Jr.,* and Sidney Goldent Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254 (Received: August 24, 1983; In Final Form: January 15, 1984)
Optical absorption spectra of solvated electrons in ammonia, methylamine, and three of their mixtures have been obtained at three different temperatures by extrapolating spectra of dilute lithium solutions to infinite dilution in metal. The solvated electron spectra in the pure solvents and in each of the mixed solvents display shape stability as temperature is changed. Values of half-height width, position of maximum absorbance,and its temperature coefficient in the mixed solvents are between the values of the corresponding quantities in the two pure solvents. Application of a two-absorber model analysis, factor analysis, and also the Gram-Schmidt procedure show that all of the solvated electron spectra in mixtures of ammonia and methylamine are representable in terms of a small number of linearly independent absorption bands as long as these absorption bands are presumed to shift with changing conditions, Le., changing temperature and/or solvent composition. For all spectra at different temperatures for a fixed solvent composition the number of such shifted absorption bands turns out to be just one. For all spectra at both different temperatures and different solvent compositions the number of such shifted absorption bands does not exceed three.
Introduction Historically, there have been many models proposed to explain the experimental behavior exhibited by the solvated electron.’ Since the mid-l940’s, when Ogg published his F-center-type structure for the solvated electron,2 what has become known as the “cavity model” has been incorporated into virtually all subsequent equilibrium models of metal solutions. The cavity model Professor Emeritus.
0022-365418412088-3804$01.50/0
presupposes that electrons are localized in solvent-free regions (cavities) in the liquid. An enormous amount of theoretical work has centered on these presupposed ~avities.~Although intuitively (1) See, for example, proceedings from Collcque Weyl I-V and references therein. (2) Ogg, R.A. J. Am. Chem. SOC.1946, 68, 155. (3) Jortner, J. J . Chem. Phys. 1959, 30, 839. Copeland, D. A.; Kestner, N. R.; Jortner, J. J. Chem. Phys. 1970,53, 1189. Brodsky, A. M.; Tsarevsky, A. V. Adu. Chem. Phys. 1980, 44,483.
0 1984 American Chemical Society
Solvated Electron Spectra in NH3 and CH3NH2 attractive, the cavity model has not been able to account for all the experimental data4 and has provided little predictive insight into the nature of these solutions. In particular, the cavity model fails to predict correctly the optical absorption spectra of solvated electrons. Comparisons between the experimental line shapes observed for solvated electron spectra and the line shapes calculated by the cavity model4 are unsatisfactory. Further refinements in the calculations only serve to worsen the situation: the line width of the calculated spectrum is too narrow, the band profile is too symmetrical, the position of the maximum is incorrect, and the thermal behavior is not correctly p r e d i ~ t e d . ~ Nevertheless, the spectroscopic behavior of the solvated electron in mixed polar solvents has been taken as evidence favoring the cavity model. These systems exhibit a single band assignable to the solvated electron, with no hint of resolution into more than one component. It has been asserted that “in mixed ammonia and methylamine ... [a] band could of course be produced from the sum of two separate bands each representing the spectrum observed in the pure solvents, but such a band would be wider than either of the two parent bands.”6 The lack of this anticipated line broadening in mixed systems has been taken as prima facie evidence that the observed spectra cannot arise from a superposition of a pair of overlapping bands, In another study, it was demonstrated that solvated electron spectra in some mixtures could not be accurately represented as a linear combination of a small number of bands each of which is fixed in position.’ In contrast to this line of development, a new model has recently been suggested to account for solvated electron absorption spectra in mixed polar solvents.* This model has been successful in accounting for the compositional behavior of half-height widths of solvated electron absorption bands in a number of polar solvent mixtures. This success indicates that the failures encountered in the earlier investigations in representing the observed spectra as linear combinations of a small number of absorption bands stemmed from the particular assumptions adopted, rather than from any fundamental inability to do so. In the first instance, the asserted line broadening was simply incorrectly anticipated. In the second instance, the assumed fixed positions of the individual components appear to have been unnecessarily restrictive. The two-absorber model used here applies to binary polar mixtures where there is no chemical reaction between the two solvents, and when the absorption band in a mixture is an experimentally unresolvable composite of two absorption bands. It is based on the following assumptions: (1) each solvated electron is localized by a solvent molecule, or cluster of molecules, to form an anionic cluster; (2) the solvent clusters are not mixed, i.e., the solvated electron is associated with one solvent or the other; (3) each solvated electron has an absorption spectrum whose characteristic feature is its line shape, but the position of the maximum may shift with changes in thermodynamic conditions; (4) the two chemically distinct solvated electrons are in chemical equilibrium with each other and with the solvents they involve. In spite of the cited success of this model in accounting for all the half-height width data available for solvated electron spectra, tests of the model using complete spectra are required. These are accomplished by using optical absorption spectra of dilute lithium metal solutions in ammonia, methylamine, and three mixtures of ammonia and methylamine.
Experimental Section Ammonia (99.99% pure) and methylamine were obtained from Matheson Chemical Co. The lithium used was from lithium Corporation of America, in the form of a wire stored under mineral oil. (4) Kestner, N. R. In “Electron-Solvent and Anion-Solvent Interactions”, Kevan, L., Webster, B. C., Ed.; Elsevier: New York, 1976. (5) Webster, B. C.; Carmichael, 1. C. J . Chem. Phys. 1978, 68, 4086. (6) Blades, H.; Hodgins, J. W. Can. J . Chem. 1955, 33, 411. (7) Dye, J. L.; DeBacker, M. G.; Dorfman, L. M. J . Chern. Phys. 1970, 52, 6251. (8) Golden, S.; Tuttle, T. R., Jr. J . Phys. Chern. 1978, 82, 944.
The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 3805 BALL
JOINT
---A N E E M E VALVE
+
OPTICAL SIDEARM
t46
CM+
1-IlZ.31 wn*(ocwi%
7
WELL
Figure 1. Apparatus for preparing dilute lithium solutions and measuring their spectra.
The ammonia was used without further purification except as occurred in the standard treatment of the solvents, as described below. The methylamine had been specially treated to remove most of the ammonia (approximately 1%) which typically contaminates the commercial p r o d ~ c t . ~(All percentages quoted are mole percent.) The lithium wire was washed twice with toluene and mechanically cleaned before being introduced into the metal purification compartment of the apparatus shown in Figure 1. The design of the apparatus used in obtaining optical absorption spectra was very similar to that used in previous studies carried out in this laboratory.1° However, two minor changes were adopted. The first, of a precautionary nature, was the insertion of a coarse fritted disk into the neck of the apparatus below the graded quartz-to-Pyrex seal. The graded seal served to attach the Pyrex access valve (Fisher-Porter 795-1 20-0004) to the apparatus. This additional fritted disk barred contact between metal solutions and the Pyrex. Spectra taken with an apparatus without this added feature did sometimes show signs of an absorption band attributable to dissolved sodium.” The second change was designed to provide a means of lithium purification in situ. It consisted of a conveniently oriented side arm which contained a medium-fritted quartz disk through which a solution of lithium metal could be filtered, a storage compartment into which the filtered solution could be deposited, and a constriction at which the purification compartment could be sealed off and removed (once the metal solution had been filtered). The first step in preparing the sample involved the treatment of the lithium to remove impurities insoluble in ammonia. A convenient amount of predried ammonia was distilled into the purification compartment containing the impure lithium. The resulting solution was forced through the medium-fritted disk into the storage compartment. The ammonia was pumped out of the apparatus leaving behind silver-colored lithium crystals. At this juncture it turned out to be important that no lithium metal crystals remained on or near the constriction, because reaction between the metal and the quartz was induced by the heating required to seal off the side arm. Any metal detected in the critical region was washed into the storage compartment by condensing ammonia onto it. The constriction was then sealed, and the purification compartment and the medium-fritted disk were removed from the apparatus. Although chemical analyses were not done in order to document any purification of the lithium which may accompany the described procedure, the solutions prepared with the treated lithium (9) Strong, J. Thesis, Brandeis University, 1970. (10) Hurley, I.; Tuttle, T. R., Jr.; Golden, S. In “Metal Ammonia Solutions, Colloque Weyl 11”,Lagowski, J. J., Sienko, M. J., Ed.; Butterworths: London, 1970. (11) Hurley, I.; Tuttle, T. R., Jr.; Golden, S. J . Chem. Phys. 1968, 48, 2818.
3806
-
The Journal of Physical Chemistry, Vol. 88, No. 17. 1984
were adequately stable for determining accurate spectra even at -30 OC (the highest temperature at which measurements were taken). As an additional indication of the beneficial effect of the treatment, lithium so treated appeared to dissolve in tetrahydrofuran and 1,Zdimethoxethane to form light blue solutions. N o such blue solutions were produced with untreated lithium metal. Dissolution of lithium metal in ethers has apparently not been reported previously. After the lithium was purified, a typical sample preparation proceeded as follows. Appropriate amounts of the two solvents, determined by measuring the volumes of the gases at 1-atm pressure, or sometimes at lower pressures, were condensed onto a Na-K alloy. The resulting solution of solvent and alloy were then subjected to several freeze-thaw cycles before the solvent was distilled into the apparatus, which itself had been flamed out in the usual manner. The solvent mixtures employed in the present study contained 1:1, 4:1, and 1:4 mole ratios of ammonia to methylamine. In what follows, these mixtures are referred to as solvent X, solvent Y, and solvent Z, respectively. Optical cells of nominal path lengths, 1 cm and 1 mm, constructed of Infrasil optical grade quartz from Precision Cells, Inc. (Hicksville, NY) were used in obtaining the absorption spectra of the lithium solutions. Cell path lengths were calibrated by measuring the absorbance of a standard copper sulfate solution. All absorbance measurements were obtained on a Cary 14R spectrophotometer operating in its IR mode.'* The spectrophotometer was interfaced to an LSI-11 minicomputer, thereby facilitating data collection and subsequent analysis. The absorbance scale of the Cary was calibrated with a standard C u S 0 4 solution. The wavelength scale was calibrated with a holmium oxide standard (Corning No. 3130): The wavelength range over which data was collected extended from about 2100 to 400 nm. A potentiometer (Beckman/Helipot Corp., Newport Beach, CA) with a 15-V bias from a power supply (Lambda Electronics Corp., Melville, NY) was attached to the Cary pen drive mechanism, generating a signal to be sent to the computer. While the spectrum of a sample was being measured, the signal from the potentiometer was sampled every second. The sampling frequency was determined by a real time clock (Data Translation, DT 2769), digitized by a 12-bit analog-to-digital converter (Data Translation, D T 2762), and stored on a floppy disk. Since the rate at which the spectra were scanned was always 2.5 nm/s, absorbances were recorded at 2.5-nm intervals throughout a spectrum. This resulted in a sample spectrum data file composed of between 500 to 700 data points, the exact number depending on the extent of solvent absorption. Base lines were recorded similarly. The absorbances were obtained by subtracting the average base line from the sample spectrum. A plot of voltage from the potentiometer vs. corresponding absorbances read from the Cary strip-chart recorder calibrated the potential signals in absorbance units. Spectra for each sample were obtained a t three temperatures: at -90, -70, and -50 OC for pure methylamine, and mixtures X and Z; at -70, -50, and -30 OC for pure ammonia and mixture Y. These latter solvents were frozen at -90 OC, precluding measurements at this temperature. The remaining details of the procedure used here were essentially the same as those reported earlier.'* Data that were not reproducible were discarded. A total of 262 full spectra were accepted for the analysis presented here.
Solvated Electron Spectra Determination. In all instances, the absorption spectra obtained for dilute lithium solutions consisted of a single featureless, asymmetric band with a long high-frequency tail. In a given solvent, at a given temperature, the spectra observed a t different metal concentrations differed very little from one another. However, at the lower temperatures, in each solvent except pure methylamine, the spectra of the more concentrated metal solutions were slightly narrower and slightly shifted to lower frequencies. For example, the breadth of the band in solvent Z at -90 O C varied from 4400 to 3900 cm-', and the position of the maximum from (12) Rubinstein, G. Thesis, Brandeis University, 1973.
.
btupak et a!.
1.0-
0.8-&6
-0.4-_ 0.2-
v
JI0JCM-j
Figure 2. Spectra of lithium solutions in liquid ammonia extrapolated to infinite dilution in metal. Relative absorbance is plotted vs. photon
frequency.
(IO'CM-') Figure 3. Spectra of lithium solutions in methylamine extrapolated to infinite dilution in metal. Relative absorbance is plotted vs. photon
frequency.
1
I
; . a
..
t 3 [/oJCM-l>
Figure 4. Spectra of lithium solutions in 50% NH3-50%CH,NH2 extrapolated to infinite dilution. Relative absorbance is plotted vs. photon
frequency. 8500 to 8100 cm-' in going from the most dilute to the most concentrated metal solutions. These small changes are similar in magnitude to corresponding changes observed for spectra of sodium solutions in ammonia at low temperatures.12 In order to remove the effects of changing metal concentration on the spectra, F(v)= A ( v ) / A ( s )(P is the frequency of maximum absorption) was extrapolated vs. A(P)to infinite dilution in metal. Values of F(v) a t infinite dilution were the intercepts of linear least-squares fits of F(v)vs. A(2). In some cases, extrapolations of F(v)to infinite dilution were obtained with a quadratic function of A ( s ) . The results of such extrapolations were identical within experimental uncertainty with the results of the corresponding extrapolations using a linear function. This insensitivity of the extrapolated value F(v) is similar to that previously encountered with dilute sodium solutions in ammonia.12 The solvated electron spectra at infinite dilution are plotted in Figures 2-6. The spectral parameters for the absorption bands
The Journal of Physical Chemistry, Vol. 88, No. 1 7 , 1984 3807
Solvated Electron Spectra in NH3 and CH3NH2
2v 1
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, I
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Figure 5. Spectra of lithium solutions in 80% NH3-20% CH3NH2 extrapolated to infinite dilution. Relative absorbance is plotted vs. photon frequency.
...
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.
.e
3
Figure 7. Demonstration of shape stability of solvated electron optical absorption spectra in ammonia at different temperatures: 0 , -70 "C; m, -50 "C; 0, -30 "C. Relative absorbance is plotted vs. reference frequency.
0.8
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20
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Figure 6. Spectra of lithium solutions in 20% NH3-80% CH3NH2 extrapolated to infinite dilution. Relative absorbance is plotted vs. photon frequency. TABLE I: Parameters Pertaining to Solvated Electron Absorption Bands and Their Shape Stability solvent P pb W,,2c Ad Ave ff "3
CH3NH2
0
-70 -50 -30
7238 6775 6430
3221 3228 3300
4086 4018 4140
451 843
1.000 1.003 0.995
-90 "-70 -50
8606 8022 7500
4715 4733 4768
6737 6698 6924
0 541 1039
1.000 1.000 0.995
-90 -70 -50
8250 7877 7395
3844 3870 3859
4968 5021 5060
0
371 807
1.000 1.000 0.996
-70 -50 -30
7643 7201 6785
3500 3520 3444
4469 5517 4600
0 428 769
1.000 1.000 0.998
-90 -70 -50
8400 7974 7557
4220 4197 4111
5868 5519 5491
0 423 794
1.000 1.020 1.019
Figure 8. Demonstration of shape stability of solvated electron optical absorption spectra in methylamine at different temperatures: 0 , -90 "C; X, -70 "C; 0,-50 OC. Relative absorbance is plotted vs. reference frequency.
.
.
. I
I
i
X
Y Z
OTemperature in " C . bFrequencyof maximum absorbance in cm-I. 'Half-height width in cm-l. dArea in cm-'. eFrequency shifts obtained from minimizing rms deviations between reference spectrum and shifted spectra. f Multiplicative factor used in minimizing the rms deviations between shifted and reference spectra. are given in Table I. The average temperature coefficients of the solvated electron bands are 20 f 3 in NH,, 28 f 2 in CH3NH2, 21 f 3 in X, 21 f 1 in Y , and 21.1 f 0.3 in Z, all in cm-'/"C. Shape Stability. Visual evidence of shape stability of solvated electron optical absorption bands in each of the five different solvents is given in Figures 7-1 1. In each solvent the lowest temperature spectrum served as reference, and the higher temperature spectra were shifted in frequency so as to minimize the root-mean-squared (rms) differences in absorption between the reference spectrum and the shifted spectra. This procedure for testing spectra shape stability differs slightly from an earlier one,l3
+
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4
6
8
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26
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Figure 9. Demonstration of shape stability of solvated electron optical absorption spectra in mixture X at different temperatures: 0 , -90 "C; 0,-70 OC; X -50 OC. Relative absorbance is plotted vs. reference frequency. although the shifts, and hence the actual comparisons, obtained from the two different methods differ very little. An additional minimization of the rms differences in absorption is obtained by varying the height of the shifted spectrum with respect to the reference spectrum. The quantities characterizing these tests of shape stability are collected in Table 11. For ammonia, the rms deviation from the average is 0.24%;for methylamine it is 0.48%; for X, 0.40%;for Y , 0.45%;and for Z, 0.78%. Clearly, shape stability is maintained for the solvated electron spectra in the pure solvents and in the mixed solvents as well. (13) Tuttle, T. R., Jr.; Golden, S. J . Chem. SOC.,Faraday Trans. 2 1981, 77, 873. Stupak, C. M.; Tuttle, T. R., Jr.; Golden, S. J . Phys. Chem. 1982, 86, 327.
3808 The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 I
'
"
"
"
'
.
'
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Stupak et al.
I 1.0 -.
+ I
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4
Figure 10. Demonstration of shape stability of solvated electron optical absorption spectra in mixture Y at different temperatures: 0 , -70 OC; 0,-50 O C ; X, -30 OC. Relative absorbance is plotted vs. reference frequency.
Tc
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Figure 11. Demonstration of shape stability of solvated electron optical absorption spectra in mixture Z at different temperatures: 0 , -90 O C ; 0,-70 OC; X, -50 OC. Relative absorbance is plotted vs. reference frequency. TABLE 11: Results from Two-Absorber Model Analysis on Spectra at -70 and -50 O C 0 YJ' Xjd C J K ~Ad fg rms* % D' -70 -70 -70 -70 -70
1.00 0.80 0.50 0.20 0.00
0.666 0.473 0.217
-50 -50 -50 50 -50
1.00 0.80 0.50 0.20 0.00
0.640 0.489 0.264
6
8
10
12
3
3*(103C M - 9
1.95 1.11 0.90
0 86 253 421 784
1.033 1.025 1.016
0.00415 0.00428 0.00274
2.29 2.15 1.22
2.25 2.95 0.70
0 -13 188 338 726
1.035 1.031 1.020
0.00327 0.00365 0.00268
1.80 1.90 1.21
OSolvent J of J-K mixtures is CH3NH,. Solvent K is ammonia. bTemperature in OC. 'Mole fraction of solvent J. dMole fraction of solvated electron J. eConcentrationproduct for reaction S, + SK= SJ + SK-,that is, CJK= x~J/xJYK./Frequency shift of experimentally determined solvated electron spectra in wavenumbers used in TAM analysis. g Multiplicative factor used to minimize the rms difference between the linear combination of pure solvent spectra and the experimental spectrum in the mixed solvent. *rms deviation between the best fit and the experimental spectrum. 'Gives the rms as a percentage of maximum absorbance.
Solvent Dependence. The variation of solvated electron absorption bands with changing solvent composition in ammoniamethylamine mixtures at -70 O C is illustrated in Figure 12. The various spectral parameters are given in Table I. At a fixed temperature, the half-height width of the band increases monotonically as the fraction of ammonia decreases. The half-height widths of the bands in the mixtures are all between the two pure solvent values. Generally, values of the frequency at the maximum, 8, also increase monotonically and are between the two pure solvent values as ammonia fraction decreases. (The value of b at -50 O C
14
16
18
20
22
24
(/O'CMY
Figure 12. Unit height normalized solvated electron spectra in mixtures of ammonia and methylamine at -70 "C: 0,NH3; A, 80% NH3-20% CHSNH2; X, 50% NH3-50% CH3NH2; 0, 20% "340% CH3NHZ; 0 , CH3 N H 2. for mixture Z is possibly an exception in this regard.) The shapes of the unit height normalized spectra of Figure 12 appear to change monotonically from the shape exhibited in one pure solvent to that in the other pure solvent. Two-Absorber Model Analysis The measured solvated electron optical absorption spectra in ammonia, methylamine, and their mixtures appear to satisfy certain conditions and requirements of the two-absorber model. Spectra in each pure solvent exhibit shape stability, and the widths and shapes of the mixed solvent spectra are between those of the pure solvent spectra. Although not directly verifiable, the postulated shape stability of the individual solvated electron bands in the solvent mixtures will derive strong indirect support by a success of the model in providing a quantitative account of the observed spectral variations. The two-absorber model (TAM) analysis was developed to fit each experimentally observed mixed solvent spectrum as a linear combination of the shifted pure solvent spectra. In practice, linear combinations were constructed from the shifted methylamine and unshifted ammonia spectra so that t ~ a I c d ( ~= ) x;tJ(v - 3K - 6v) + xitK(V - 3,) (l) in which J and K denote methylamine and ammonia, respectively, the ['s are unit area normalized spectra, D~ is the frequency of maximum absorption of the ammonia spectrum, 6 v is the frequency of displacement between the maxima of the shifted methylamine and unshifted ammonia spectra, and the x"s are the fractional contributions of the two spectra. A best fit of the calculated spectrum to the experimental mixed solvent spectrum was obtained by shifting the area-normalized mixed solvent spectrum, by Av, in order to minimize (&,,,(v + Av) - fcalcd(u))*. When [calcd(v) = &,,lx(v + Av) within experimental uncertainty the X'S of eq 1 were assumed to be the electron mole fractions (unprimed x's) determined by
in which the unit height normalized spectra, the 4's, and their areas, A's, as well as the shifts, 6 v and Av, are all known. As a result, specifying values of 6 v and Av in the fitting procedure automatically determined the values of xJ and xKbecause of eq 2 a n d x J + X K = 1. The results of applying the T A M analysis to the data at -50 and -70 O C with 6 v = 0 are given in Table 11. The rms deviations between experimental spectra and their best fits do not exceed 2.29% of the maximum absorption and average 1.78%. Detailed comparison of the experimental spectra in mixtures X, Y , and Z at -70 O C with there best fits are presented in Figures 13-1 5. Results of applying the TAM analysis to data at -70 O C with 6v values ranging from -1000 cm-l to 1000 cm-' are summarized in Table 111. It is remarkable that neither the quality of the fits nor the value of the composition variable are much affected over
The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 3809
Solvated Electron Spectra in N H 3 and CH3NH2
0.20.-
G 0.12
I
*
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.
a
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8
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Figure 13. Comparison of solvated electron absorption spectrum in 50% NH3-50% CH3NH2at -70 OC with best fit from two-absorber model analysis: 0 , experimental; X, best fit. Area normalized absorbance is
plotted vs. frequency.
TABLE 111: Effect of Displacing the Maxima of Pure Solvent Solvated Electron Bands on Two-Absorber Model Fits % D' 6Va rmsb XJd 1000 0.007 46 3.76 0.422 2.57 500 0.46 1 0.005 11 0.468 400 2.49 0.004 96 2.39 0.004 77 0.475 300 0.473 2.26 0.004 51 200 0.476 2.17 0.004 32 100 2.15 0.004 28 0.473 0 2.45 0.004 88 -100 0.47 1 2.31 -200 0.467 0.004 60 -300 0.462 2.34 0.004 66 0.457 2.37 -400 0.004 73 2.34 -500 0.450 0.004 66 0.398 -1000 2.62 0.005 22 a Displacement of maxima of pure solvent solvated electron bands. bRoot-mean-squareddeviations of the linear combinations of pure solvent spectra from the experimentally determined solvated electron spectrum in mixture X at -70 "C. cDeviation as a percent of the maximum absorption. Fraction of methylamine solvated electron.
TABLE IV: Results of Factor Analysis of Shifted Spectra at -50 and -70 OC EO rms* %De Ea rmsb % DC T -70 'C T = -50 OC 4.70 30.5489 0.0457 4.57 30.5954 0.0470 0.0148 1.48 0.2960 0.0108 1.08 0.2981 0.0043 0.43 0.0107 0.0068 0.68 0,0299 0.18 0.0062 0.0021 0.21 0.0024 0.0018 0 0 0.0007 0 0 0.0005
0.04
Principal eigenvalues. Rms deviation between experimental spectra and best fits obtained by using the first n-principle eigenvectors. Rms as a percentage of the maximum.
t . t
0.06 0.04
0
1
. I
I
I
, V
(10'
4
.-
'.'
t t t t tJ
CM-I)
Figure 15. Comparison of solvated electron absorption spectrum in 20% ",-SO% CH3NH2at -70 "C with best fit from two-absorber model
analysis: 0 , experimental; X, best fit. Area normalized absorbance is plotted vs. frequency.
a substantial range of 6v values. Similar results were obtained for the -50 "C T A M analysis.
Supplementary Analysis In order to provide independent tests of the results obtained from the TAM analysis, principal factor analysis and the GramSchmidt procedure were applied to appropriately shifted spectra at -50 and -70 OC. In each case five spectra, one from each solvent at a given temperature, were used. The shifts can be derived from Table I. Principal factor analysis has been employed extensively in determining the number of different absorbers contributing to a set of experimentally determined spectra. The details of the method are given elsewhereI4 and will not be repeated here.
Essentially, this method casts the analyses into the form of an eigenvalue problem in which the number of nonvanishing eigenvalues of the so-called correlation matrix is identified as the number of independent absorbers. The results of applying principal factor analysis to the appropriately shifted spectra at -50 and -70 O C are given in Table IV. Listed in the table are the magnitudes of the principal eigenvectors and the rms deviations between the experimental spectra and the best fits obtained by using the first n-principal eigenvectors as a basis. Clearly, the best fits using only a single principal eigenvector are inadequate at both temperatures, as judged by the corresponding rms deviations. With two principal eigenvectors, the rms deviations of the best fits of the data are considerably reduced. With three principal eigenvectors the deviations are well within experimental errors. The Gram-Schmidt orthogonalization procedure is a wellknown method of constructing orthogonal functions from a set of linearly independent mathematical function^.'^ It can be shown that, if the members of a set of functions are not linearly independent, the Gram-Schmidt process will produce only a number of orthogonal functions equal to the number of linearly independent members of the set. It is this property that may be exploited to enumerate the linearly independent spectra among a given set of spectra. In the implementation of the Gram-Schmidt process each shifted spectrum is a vector whose components are absorptions at 500-cm-' intervals in the frequency range of 5.5 X lo3to 20.0 X lo3 cm-I. With this identification, application of the GramSchmidt procedure to the appropriately shifted spectra at -50 and -70 OC yielded the results given in Table V. Judged by the rms deviations of the fits, a basis set of one is deemed inadequate to reproduce the data. For a basis set of two, the fits are much improved, although possibly not within experimental uncertainty. With a basis set of three, the fits are again improved and now appear to lie well within experimental uncertainty. (14) Malinowski, E. R.; Howery, D. G. "Factor Analysis in Chemistry"; Wiley-Interscience: New York, 1980. (1 5 ) Birkhoff, G.; Maclane, S. "A Survey of Modern Algebra"; Macmillan: New York, 1953.
3810 The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 TABLE V: Results of Applying Gram-Schmidt Procedure to Shifted Solvated Electron Spectra at -50 and -70 ‘C’ basisb Jc KC XC YC ZC T = -70 O C 1 0 0.1144 0.0816 0.1009 0.0592 0.0275 2 0 0 0.0367 0.0220 0.0108 3 0 0 0 0.0050 0.0108 4 0 0 0 0 5 0 0 0 0 0
1 2 3 4 5
0 0 0 0 0
T = -50 OC 0.1148 0.0809 0 0.0220 0 0 0 0 0 0
0.0993 0.0197 0.0086 0 0
0.0609 0.0181 0.0144 0.0119 0
“Numbers in table are rms deviations of fits obtained by using the number of basis functions indicated in the first column. To convert the rms to a percentage of the maximum, multiply by 100%. bNumber of basis functions. The basis functions used are the experimental spectra, and are used in the same order as the solvents are listed in the top row, from left to right. CSolvent: J is CH3NH2,K is NH3. X, Y, and Z are 1:1, 1:4,and 4:l J to K mole ratio.
Discussion The results obtained with principal factor analysis and/or the Gram-Schmidt process on the appropriately shifted spectra (see Tables 11, IV and V) at either -50 or -70 OC are essentially the same as the results obtained with the two-absorber model analysis. At least two independent absorbers are required to account for the observed spectra. In addition, the factor analysis and Gram-Schmidt results indicate that no more than three independent absorbers are required to account for the observed band shapes. These conclusions are in sharp contrast to the conclusion of earlier investigator^.^ For solvated electron optical spectra in mixtures of ammonia and methylamine it appears that the two-absorber model comes close to providing the detailed, quantitative accounting which we seek of the compositional behavior of the band shape. However, because the deviations between the experimental spectra and their best fits exceed slightly the deviations anticipated on the basis of our estimate of experimental uncertainty of 0.5% the question of the possible source of these unanticipated deviations arises. One possibility is that the experimental uncertainty has been estimated at too low a value. This seems unlikely in view of the large number of spectra raken, and the excellent reproducibility obtained. Another possibility is one that has already been mentioned, Le., that the spectra in the mixed solvents actually arise from more than two absorbers. A third possibility is that any rigorous fitting of the mixture data by a two-absorber model must allow for slight changes of up to a few percent in the spectral shapes attributed to the individual absorbers. A fourth possibility is that the implicit assumption that the frequency dependence of the refractive indices of the pure solvents and of the mixtures affect the shapes of the bands inappreciably is not correct. However, no large effect due to the frequency-dependent refractive indices should be anticipated, because they are undoubtedly slowly varying functions of frequency in the region where solvated electrons absorb appreciably. Unfortunately, no definitive resolution of this point is possible a t present due to lack of pertinent refractive index data. The description of solvated electrons in binary mixed solvents given here appears to differ markedly from one previously accorded such electron^.^ The principal difference seems to be that here the electrons are imagined to be localized about specific sites which can serve to distinguish electrons from one another. The earlier view, in contrast, imagines the electron to lie “...delocalized so that the electron can sample the average environment of the mixture...”’ Nevertheless, the extent to which the electron is localized can hardly be at issue here because its dispersion in position can be determined directly from the optical absorption spectrurn.l6 Rather, the question seems to be what character these
Stupak et al. sites impart to the excess electrons. According to the earlier view, the solvated electron optical spectra reflect the average composition of the mixed solvent, while the view expressed here is that the spectra reflect distinctly different properties for electrons at different sites. An important feature of the analyses carried out here is the shifting of the individual solvated electron absorption bands which has the effect of reducing the number of functions required to reproduce the data within experimental uncertainty. For example, the application of factor analysis to unshifted solvated electron spectra in a pure solvent (see Figures 2 and 3) yields a number of absorbing species greater than one, even though the actual number of absorbing species there is universally accepted as being one. Clearly, appropriate shifting of the absorption bands is required in the pure solvents before the analysis can yield the correct number of absorbing species. As a matter of consistency this shifting has been assumed also to occur for the individual solvated electron bands in the mixtures. Even though the previous application of the two-absorber model analysis in accounting for the compositional behavior of half-height widths of solvated electron optical absorption bands8 has been criticized for “...not dealing with the shift of the absorption bands of the two solvent anions to nearly the same position in mixtures even though the anions absorb at different positions in respective pure liquids”,17 in the earlier works it was pointed out that the energetic constraint imposed by the chemical equilibrium between the two anions leads to the virtual superposition of their absorption bands. Such shifting is universally regarded as a medium effect to be attributed to the changing surroundings of an ion. Analysis of incorrectly shifted spectra, including unshifted spectra, can only lead to overestimating the number of absorbers. Another result of the two-absorber model analysis are the values of the concentration product, CJK,for the equilibrium between the two anionic complexes. For the ammonia-methylamine system studied here the product has a value at -70 OC of 1.33 f 0.42 and at -50 ‘C of 1.30 f 0.63. Clearly, neither solvent anionic complex is greatly favored over the other, although the ammonia species may be slightly favored at both temperatures. Also, within experimental uncertainty, the free energy, enthalpy, and entropy changes for the reaction between these two different electrons are all zero. More accurate evaluation of these quantities requires knowledge of the relevant solvent activities. Nevertheless, the apparent lack of shifting of the equilibrium between the solvent anions as temperature changes suggests that the observed shape stability of the optical absorption bands in each mixed solvent as the temperature is changed results from the proportions of the two absorbers remaining fixed as well as their individual band shapes. It might also seem as though the relative positions of the two-component band would have to remain constant in order to ensure shape stability. However, since we can hardly tell the difference between a composite band with a relative shift of +500 cm-’ from one with a relative shift of -500 cm-I (see Table 111), the possibility that the two-component bands have substantially different temperature coefficients cannot be discounted. In particular, whether the two-component bands have identical temperature coefficients equal to that observed experimentally, or whether each component band has a temperature coefficient characteristic of its solvent (-20 cm-’/K for ammonia and -28 cm-’/K for methylamine) giving rise to a weighted average temperature coefficient for the experimental band cannot be distinguished with the available data.
Acknowledgment. This project was supported in part by BRSG SO7 RR07044 awarded by the Biomedical Research Support Grant Program, Division of Research Resources, National Institutes of Health. Registry No. NH3, 7664-41-7; CH3NH,, 74-89-5. (16) Golden, S.; Tuttle, T. R., Jr. J. Chem. SOC., Faraday Trans. 2 1979, 75, 414. (17) Discussion, J . Phys. Chem. 1980, 84, 1139.
