J. Phys. Chem. 1981, 85, 856-861
858
Metal-Ammonia Solutions. 14. Electron Spin Resonance at -65
OC
R. L. Harris7 and J. J. Lagowski" Welch Chemical Laboratory, The University of Texas, Austin, Texas 78712 (Received: May 8, 1980)
ESR measurements were conducted on sodium-, potassium-, rubidium-, and cesium-ammonia solutions at -65 "C in the concentration range of M metal to M metal in an attempt to obtain a single set of self-consistent data. The g value, spin-spin and spin-lattice relaxation times, and concentration of paramagnetic species were measured. The data for the g values and concentration of paramagnetic species as a function of metal concentration suggest that the species in solution are the solvated electron, e-, and the solvated electron-metal cation ion pair. e-M+. The electron spin-spin and spin-lattice relaxation times are not equal at low metal concentrations.
Introduction Metal-ammonia solutions have been widely studied by use of visible spectros~opy'-~ and electron spin resonance (ESR) spectroscopy.'*% Unfortunately, the reported ESR experiments have been performed a t temperatures different from those at which the near-IR spectral experiments are reported; accordingly, comparison of the results obtained from these two types of experiments is difficult. Since many properties of metal-ammonia solutions appear to change markedly with changes in temperature and concentration,26meaningful comparison of the results arising from several techniques can be had only at the same temperature and metal concentration. This investigation was undertaken primarily to collect ESR data at the same temperature (-65 "C) for which the greater portion of the near-IR data is available. Secondarily, we hoped to obtain information on the equilibria present in these solutions and possibly to obtain stoichiometric data for the ammoniated electron a t this temperature. The g values of metal-ammonia solutions have been measured by a number of worker^,^^-'^^^^ and these values are shown in Table I. Both the Gouy balance te~hnique~'-~l and the ESR t e c h n i q ~ e ~ can ~ p ~be~ used - ~ ~ to obtain the total concentration of paramagnetic species. Figure la-c (parts a and b are found in the supplementary material; see paragraph at end of text regarding supplementary material) shows data obtained for metal-ammonia solutions at 25.0, 0.0, and -33.0 "C; the data available prior to this work are sparse at lower temperatures. Also, very few data have been obtained on alkali metals other than sodium and potassium. Rubinstein has observed that the spin susceptibility appears to be related to the shift of the near-IR band maxima with increasing c~ncentration.~~ Therefore it was important for us to investigate the spin susceptibility for several different alkali metals dissolved in liquid ammonia at -65 "C, the temperature a t which the most extensive collection of near-IR data is The theory now most widely accepted for the existent ESR data interprets the narrow absorption lines observed in terms of the motional narrowing of the nitrogen nuclei spin interaction with the solvated electron pins.^',^^ The equations developed to explain the narrowness of the ESR absorption lines are l/T1 = 2.4 X 10-31(P2/N)(~c/(1 + 7 2 ~ 2 ) ) (1)
TABLE I: Reported g Values for Metal-Ammonia Solutions metal
Li Na K
Rb
cs
Ca Ba Sr
Eu
25 "C 2.0006 (2)' 2.0004 (2)' 2.0006 2.0012 2.0004 2.0002
(2)' (2)' (2)' (2)'
g value -33 "C 2.0008 ( 2)' 2.0008 (2)' 2.00089 ( 3 ) b 2.0009 ( 2)' 2.00089 (2)b 2.0005 (2)' 2.0005 (2)' 2.00088 ( 3 ) b
- 6 5 "C 2.00090 (3)b
2.00088 ( 7 ) b 2.00083 (6)b 2.00086
2.001 (1)' 2.001 (1)' 2.001 ( 1 ) U 2.0004 (2)' 2.0014 (2)d
a Reference 20. This work, DPPH = 2.00360. erences 1 0 and 19. Reference 24.
Ref-
laxation times, P is the square of the electron density at each nitrogen nucleus, N is the average number of such (1)R. C. Douthit and J. L. Dye, J. Am. Chem. Soc., 82,4472(1960). (2)M. Gold and W. L. Jolly, Inorg. Chem., 1, 818 (1962). (3)W. H. Koehler and J. J. Lagowski, J.Phys. Chem., 73,2329(1969). (4)G. Rubinstein, T. R. Tuttle, and S. Golden, J. Phys. Chem., 77, 2872 (1973). (5)R. K.Quinn and J. J. Lagowski, J. Phys. Chem., 73,2326 (1969). (6)P. F. Rusch and J. J. Lagowski, J. Phys. Chem., 77, 210 (1973). (7)W. Peer and J. J. Lagowski, J. Phys. Chem., 84,1110 (1980). (8)R. L. Harris and J. J. Lagowski, J. Phys. Chem., 84,1091(1980). (9)W. Peer and J. J. Lagowski, unpublished results. (10)C. A. Hutchison, Jr., and R. C. Pastor, J. Chem. Phys., 21,1959 (1953). (11)C. A. Hutchison and D. E. O'Reilly, J. Chem. Phys., 34, 1279 (1961). (12)D. Cutler and J. G. Powles, Proc. Phys. SOC.,82,1 (1963). (13)D. Cutler and J. G. Powles, Proc. Phys. SOC.,80, 130 (1962). (14)V. L. Pollak, J. Chern. Phys., 34,864 (1961). (15)D. E. O'Reilly, J. Chem. Phys., 50,4743 (1969). (16)W. S. Glaunsinger and M. J. Sienko, J. Chem. Phys., 62,1873 (1975). (17)P. Damay and M. J. Sienko, J. Phys. Chem., 79, 3000 (1975). (18)R.A. Levy, Phys. Reu., 102,31 (1956). (19)C. A. Hutchison, Jr., and R. C. Pastor, Phys. Reu., 81,282(1951). (20)R. Catterall and M. C. R. Symons, J. Chem. SOC.,4342 (1964). (21)S. I. Chan, J. A. Austin, and 0. A. Paez, "Metal Ammonia Solutions", Proceedings of Colloque Weyl 11, J. J. Lagowski and M. J. Sienko, Ed., Butterworths, London, 1970,pp 425-438. (22)R. Catterall and M. C. R. Symons, J. Chern. SOC.,3763 (1965). (23)T.David, P. Damay, and M. J. Sienko, J. Chern. Phys., 62,1526 (197.5). . -,. \--
1/T2 = 1.2
X
1 0 - 3 1 ( p / N ) ( ~+ c ~ ~ /+(7,2w02)) 1 (2)
where T , and T 2 are the spin-lattice and spin-spin re+Bell Laboratories, 600 Mountain Ave., Murray Hill, NJ 07974.
(24)D.S. Thompson, E. E. Hazen, and J. S. Waugh, J. Chem. Phys., 44, 2954 (1966). (25)D.E. O'Reilly, J. Chem. Phys., 35, 1856 (1961). (26)J. C. Thompson, "Electrons in Liquid Ammonia", Oxford University Press, Oxford, 1976,p 80. (27)E. Huster, Ann. Phys., 33, 477 (1938).
0022-3654/81/2085-0856$01.25/00 1981 American Chemical Society
The Journal of Physical Chemistry, Vol. 85, No. 7, 1981 057
ESR of Metal-Ammonia Solutions lo"
.I6
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1 ~~~
~
ANA
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Figure 1. (a) Paramagnetic spin concentrationsfor metal-ammonia solutions at -33 O C . Shadings indicate authors. Symbols indicate metal used. The line indicates the expected spin pairing behavior for free electrons.
nuclei, T, is the correlation time of the interaction, and wo is the angular momentum of the resonance. The spin-spin relaxation time, T,,can be obtained from the width of the ESR absorption line or from pulsed ESR experiments. The spin-lattice relaxation time can be obtained either from a careful determination of the saturation behavior of the ESR absorption as the perpendicular field is increased,l@l' or directly by means of pulse relaxation studie~.l~-'~ The results of past relaxation-time studies are described in the supplement. Note that these data are incomplete, particularly at lower temperatures. Experimental Section An apparatus constructed entirely of quartz, similar to Figure 1 of ref 24 with an additional chamber where the metal was placed, permitted the simultaneous preparation of five metal-ammonia solutions. It was carefully washed in alcoholic KOH, dilute nitric acid, and then several times with distilled water. The tubes were then sealed at the ends; the apparatus was assembled and attached to a vacuum line where it was tested for leaks. It was removed from the vacuum line after a preliminary rinse in anhydrous ammonia, and placed in a drybox where the alkali metal sodium, potassium, rubidium, or cesium was installed in the sidearm of the apparatus. After reattachment to the vacuum line the apparatus was rinsed with anhydrous ammonia, and then again evacuated. The metal was distilled to the preparation chamber. Ammonia was then condensed onto the preparation chamber and into 1.0- or 2.0-mm (id) ESR tubes; the metal and ammonia mixed by the bumping of solution as the ammonia was evaporated to the desired volume. The ESR tubes were then removed from the body of the apparatus with an oxygen torch and stored in liquid nitrogen. It was discovered that incomplete mixing of the (28) S. Freed and N. Sugarman, J. Chem. Phys., 11, 354 (1943). (29) J.-P. Lelieur and P. Rigny, J. Chem. Phys., 59, 1142 (1973). (30) J.-P. Lelieur, Ph.D. Dissertation, Orsay, 1972. (31) R. G. Suchannek, S. Naiditch, and 0. J. Klejnot, J . Appl. Phys., 38, 690 (1967). (32) C. A. Hutchison, Jr., and R. C. Pastor, Rev. Mod. Phys., 25, 285 (1953). (33) A. DeMortier, M. DeBacker, and G. Lepoutre, J. Chirn. Phys., 380 (1972). (34) A. DeMortier, Ph.D. Dissertation, Lille, 1970. (35) 6.Rubinstein, J. Phys. Chem., 79, 2963 (1975).
P
.a?
.04
06
.08
.IO
.I2
AV/%
Figure 2. (a) Example of the kylstron mode reflection from a loaded cavity. (b) Plot converting % d t o % u .
metal-ammonia solutions (probably due to the small radius of the ESR tubes which was required to compensate for the poor rf penetration at the frequencies used) sometimes occurred on thawing the ESR tubes; mixing of the solutions was effected with a sonic mixer after which the tubes with visible coloring took on a uniform color throughout. Several hours were required to mix these solutions during which time some decomposition of the metal-ammonia solutions may have occurred. Three types of ESR spectra were obtained. The first type of spectrum was recorded to obtain the g value and the height of the signal. Then a series of about 20 spectra were recorded per sample a t different microwave cavity powers to obtain the width and height of the paramagnetic signal as a function of microwave power. The third type of spectrum (the oscilloscope pattern of the Varian E9 spectrometer when operating in the reflection mode) was recorded by using a Polaroid camera while the sample was in the cavity, and was obtained to enable calculation of the loaded Q of the cavity; a typical spectrum is shown in Figure 2. About one third of the tubes either were broken in handling or showed no ESR absorption (probably due to decomposition and/or low metal concentration). Special problems occur for absorptions as narrow as the ESR signals of metal-ammonia solutions prove to be. Some samples exhibit ESR absorption widths on the order of 3.0 mG. A modulation frequency of 100 kHz gives modulation side bands at 36.0 mG for the central line.36 These badly distort the shape of the signal and can only be minimized by using a lower frequency modulation. It was discovered that, even though distortion of the ESR line shape occurs a t a modulation frequency of 1.0 kHz when the line width was narrower than about 5.0 mG, the width of the signal for even the very narrowest absorptions observed was not appreciably affected. Most spectra were obtained with a modulation frequency of 10.0 kHz, which has sidebands at about 4.0 mG. Another common problem encountered with very narrow absorption lines is modulation b r ~ a d e n i n g .The ~ ~ Varian E9 spectrometer was modified to give an additional decade of modulation signal attenuation by placing an extra resistor in the circuit described by Figure 1 of ref 37. ~~
(36) J. E. Wertz and J. R. Bolton, "Electron Spin Resonance: Ele-
mentary Theory and Practice Applications", McGraw-Hill, New York, 1972, pp 452-456. (37) I. B. Goldberg, J. Magn. Reson., 32, 233 (1978).
858
The Journal of Physical Chemisfty, Vol. 85, No. 7, 1981
Harris and Lagowski
Metal-ammonia solutions also have extremely high conductivity, thus giving rise to a phenomenon known as conduction electron spin resonance (CESR)38-41which tends to distort ESR absorptions so that they are no longer classically Lorentzian. The CESR distortion can be somewhat minimized by maintaining very thin samples (allowing complete rf field penetration of the sample), however, it can not be significantly decreased except by decreasing the perpendicularfrequency (rf field), an option not available on the Varian E9 spectrometer. The metal concentrations of the ESR samples were measured by neutron activation. The details of these measurements are described in the supplementary material. Data Reduction g Values. The g value of the sample was obtained by comparison with the g value of a DPPH sample, assumed42 to be 2.00360. Paramagnetic Concentrations, The paramagnetic concentrations of the solutions were determined by comparison with the area of the DPPH absorption. The expression used for this comparison is Ns = Nrd,2h,(mao)2/(d,2hr(AHro)2)
(3)
where N , is the concentration of the sample spins in moles/liter, N , is the concentration of the reference spins in moles/liter, d, and d, are the diameters of the reference and sample ESR tubes, respectively, h, and h, are the intensities of the absorption signals for the reference and sample, respectively, and AHroand A H S O are the extrapolated widths of the DPPH and sample ESR signals at zero power. For those samples exhibiting CESR asymmetry a better value of total area was obtained by averaging intensities from the baseline in both directions (the upper and lower half of the derivative signal). Asymmetry due to scan speed was detected by scanning the sample in both directions. Corrections were made by using only the leading side of the absorption. Corrections for partial saturation of the sample were applied to all samples via the equation
M = ( 1 + P0/(2P,,,))3/2
(4)
where M is the correction factor for N , Po is the power at which the intensity h, was measure& and P,, is the power at which the absorption intensity of the sample is at a maximum. Spin-Spin Relaxation Times. The width of the ESR signal at very low microwave power gives the spin-spin relaxation time, T2. Unfortunately, at sufficiently low microwave power such that saturation is not an absorption broadening mechanism, the intensity of the signal in these experiments was too low to detect with good accuracy. Accordingly, the spin-spin relaxation times were determined by using the method described by Poole and Fara ~ hwhich , ~ ~uses
(mpp)2 = (2/(31/2?’eT2))2(1 + r3i2T1Tz)
(5)
where AHppis the measured widths (as a function of mi(38) F. J. Dyson, Phys. Rev., 98, 349 (1955). (39) G. Feher and A. F. Kip, Phys. Reu., 98, 337 (1955). (40) N. Bloembergen, J. Appl. Phys., 23, 1383 (1952). (41) W. S. Glaunsinger and M. J. Sienko, J. Magn. Reson., 10, 253 (1973). (42) R. S. Alger, “Electron Paramagnetic Resonance: Techniques and Applications”, Interscience, New York, 1968, pp 203-204. (43) C. P. Poole, Jr., and H. A. Farach, “Relaxation in Magnetic Resonance”, Academic Press, New York, 1971, pp 4-15.
016
I
1.0
2.0
1.0
4.0
P (tu)
Flgure 3. Plot of H, vs.
P used to obtain T,.
crowave power), and Y~ is the gyromagnetic ratio. The microwave power is directly proportional to the square of so a plot of P vs. (AHp )2 will have an intercept of 4/(37ZTz2)and a slope of 4 i T 1 / ( 3 T 2 ) where , k is a proportionality constant. An example of a plot of P vs. AHp: to obtain T2is shown as Figure 3. For those few samples that exhibited a vastly different k than the 1.2 average, the results were considered defective and ignored. Furthermore, some samples exhibited a curved plot, unlike the plot shown in Figure 3, and the results from those samples were likewise ignored. The curved plots are probably a result of CESR effects.44 Spin-Lattice Relaxation Times. Spin-lattice relaxation times are determined by using a derivation45of the Bloch equations. Since T2has already been determined we can substitute eq 5 into
Ti = 1/(27’2~,2H1,,2)
(6)
Ti = 31/2Mp:/(4?’3im,2)
(7)
to obtain To obtain T1,it is necessary that H1be determined; H1 may be obtained45from 2H1 = ( 4 ~ / ( a c()a)Q L P / ( b ~ ~ 2 v 0 3 ) ) 1 / 2 (8) where a, b, and c are cavity dimensions, vo is the frequency at resonance, P is the power of the microwave signal in erg/s, and QL is the loaded Q of the cavity. Equation 8 is applicable only to the TElo2type which was used for all the work reported here. QL may be obtained by use of the cavity reflection bandwidth,46using the relation QL = vo/Av (9) where vo is the microwave frequency at resonance and hv is the cavity reflection bandwidth at resonance. Substitution of eq 8 and 9 into eq 7 gives Ti = 1.44 X 10-7~02A~AHppo/P,B, (10) where vo is the frequency at resonance, Au is the bandwidth of the cavity reflection, and P,, is the microwave power in watts, at which the intensity of the sample signal is at a maximum, as shown in Figure 4. The microwave power (44) R. L. Harris, Ph.D. Dissertation, University of Texas, 1979. (45) “V-4502EPR Spectrometer Systems”, Varian Associates, Instrument Division, Palo Alto, CA, Publication No. 87-100-123, pp 5.7-5.26. (46) C. P. Poole, Jr., and H. A. Farach, “Relaxation in Magnetic Resonance”, Academic Press, New York, 1971, pp 18-28.
The Journal of Physical Chemlstry, Vol. 85, No. 7, 1981 8SQ
ESR of Metal-Ammonia Solutions
T
-33 oc
P",,
i
h
v
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CUTLER
AND
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METAL/LIlER
Flgure 6. Spin relaxation rates for metal-ammonia solutlons at -33 "C. Symbol shapes indicate metal used. Symbol shadings indicate data source and relaxation times. The error bars are representative for all samples.
"'1
2,s
I/
e
5,O
P (m)
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@ T, @ T,
A NA 0K
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v
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CUTLER A N D POWLES THISWORK
;B
cs
+ 4 1 1 1 1 b l
I
1 1 1 / ( 1 1 1
io-'
I
I
I L J
10-2
uairi kraL/L,rEa
Figure 5. Paramagnetic spin concentrations for metal-ammonia solutions at -65 O C . Symbols indicate the metal used and the line indicates the expected spin pairing behavior for free electrons. The error bars are representative for all samples.
and microwave frequency were recorded directly from the microwave bridge console. Unfortunately, the parameters Ad and d defined in Figure 2a are not directly proportional to Av and V d , where V d is the difference in frequency between one side of the cavity reflection and the other. Calibration of the cavity reflection mode was made graphically as shown in Figure 2b. The ratio Adld is compared to the ratio Avlvd. Since Ad, d , and vd can be directly measured, Av can be calculated. Results g Values. The g values are the same for all metals in ammonia at all concentrations measured. The g values so calculated for each alkali metal in ammonia are shown in Table I. Additional results to those indicated in Table I show no change in g values of potassium-ammonia solutions and rubidium-ammonia solutions between -75.0 and +25.0 "C. There was also no perceptible change of the g value in dilute metal-ammonia solutions from a microwave power of 10.0 mW (well above saturation) to a power of 0.05 mW (well below saturation). Concentration of Paramagnetic Species. The paramagnetic spin concentrations calculated as described above are presented in Figure 5, and are used in Figures 6 and 7 as the metal concentrations, since the determination of
'IOLES I I E T A L / L I T E R
Flgure 7. Spin relaxation rates for metal-ammonia solutions at -65 O C . Symbol shapes indicate metal used. Symbol shadings indicate data source and relaxation times. The error bars are representative for all samples.
the analytical metal concentrations are of low accuracy, and could not be determined for a large number of samples (see supplement material). The measured paramagnetic concentrations are tabulated in Table I11 (supplementary material). Those paramagnetic concentrations obtained at a temperature of -33.0 "C are shown in Figure la. The good correspondence with the expected values at -33.0 "C lends credence to the work reported here at -65.0 "C. It should be noted that the paramagnetic concentration results we report for -65 "C are at variance with the older results reported by Freed and Sugarman28at -55 "C and HusterZ7at -75 OC. All of the results reported to data at temperatures below -33.0 "C were obtained by the Gouy method. The Gouy method, regardless of the care taken, tends to give a large, increasing error as the solutions are made more dilute.26 Thus, we believe that although our ESR results do not agree well with those obtained previously by means of the Gouy method at slightly different temperatures, the ESR technique is more reliable for measurement of paramagnetic concentrations at these low values.
860
Harris and Lagowski
The Journal of Physical Chemistry, Vol. 85, No. 7, 1981 + DILUTE
4.
INIERMEDIATC REGION
CON;E::;;TED
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Figure 8. Variation of the near-IR band maxima with concentration. Shadings indicate authors, symbols indicate metal used.
Relaxation Times. Although above about M metal concentration the relaxation times calculated may not be the actual relaxation times due to a breakdown in the Bloch equations (which do not account for interactions involving more than one spin), Figures 6 and 7 clearly indicate that the relaxation times are not equal for metal-ammonia solutions at the concentrations measured. The data are presented in Figures 6 and 7, as well as in Table IV of the supplementary material. The error we report in the determination of T2is larger than that reported by other workers. The primary cause of this error is due to the type of instrument we used, which although good for low spin concentrations, is not ideal for extremely accurate line-width measurement, even when the data are extrapolated to zero rf power as described in the Experimental Section. The error in the determination of Tl in continuous wave experiments is always greater than T2,since in addition to line widths other factors, i.e., P-, QL,enter into the determination. The precision of the measurements can be judged from the graph; the accuracy of the measurements is estimated, and shown as sample error bars on the graph. Standard deviations are less than the expected accuracy of the measurements. Discussion Possible Species. Previous to this work there was a general belief that the variation in position of the band maximum in the near-IR spectrum with changing metal concentration (see Figure 8) was c ~ r r e l a t e dwith ~ ~the ~~~ loss of paramagnetic molar susceptibility in metal-ammonia solutions (see Figure 5). We have previously showns that the variation in the near-IR spectrum with changing concentration is due to solvated electron interaction with the metal. Metal-related species that have been proposed8 for metal-ammonia solutions are (1)the monomer, (2) the metal anion, M-; (3) the metal dimer, M,; (4)ion pairs, M+e-; and (5) ion triplets, e-M+e-. The monomer and the ion pair are predicted to be paramagnetic, and are therefore generally excluded as candidates for the species responsible for the shift in the near-IR band with changing concentration. (47) J. C. Thompson, “The Chemistry of Non-Aqueous Solvents”, Vol. 2, J. J. Lagowski, Ed.,Academic Press, New York, 1967, pp 265-317.
The spin concentration clearly mimics the analytical concentration of the metal at -65.0 “C throughout the concentration ranges for which susceptibility data were gathered. Since the data were gathered in the dilute concentration range and throughout the intermediate concentration range (see the definitions, Figure a), it is clear that there is no correlation between the electron spin pairing and the change of the near-IR band maxima with changing concentration at -65 OC. The g values observed in this work do not change (within experimental error) with increasing concentration. Paramagnetic electron g values are relatively sensitive to the environment.48 The observation that the g value does not change implies that the environment of the electron does not vary markedly over the concentration range studied. Thus, of the proposed paramagnetic species, the monomer and the ion pair, the g values strongly imply that the ion pair is the best model as the species responsible for the shift of the near-IR band maxima in the intermediate concentration range. Correlation Times. We have previously developed“ spin site-exchange equationss0 for metal-ammonia solutions, but the equation presented in ref 49 is incorrect. A correct derivation is presented in the supplementary material and gives 1/T2= l / T 1+ f M 2 ( A ~ ) 2 ~ e , (11) where fM is the fraction of electron occupation at the cation, Aw is the difference in angular frequency between the absorption of the electron at the two sites, and T,, is the exchange time of the electron. The conclusions of ref 49 remain unaffected. The electron-nitrogen nuclei correlation time (relative motion of the electron and nitrogen nuclei), T,, contains both, T ~ , ,the contribution to 7, due to the site exchange of the electron, and Td, the contribu~ ~ equation was tion to 7, due to ammonia d i f f ~ s i o n . This presented incorrectly in ref 51, but the conclusion of the paper remains unaffected. The correct equation is l / T c = l / T e x + 1/Td (12) (48) A. Carrington and A. D. McLachlan, “Introduction to Magnetic Resonance”, Harper and Row, New York, 1967, pp 141-143. (49) R. L. Harris and J. J. Lagowski, J. Chem. Phys., 67, 4782 (1977). (50) A. C. McLaughlin and J. S. Leigh, Jr., J. Mag. Reson., 9, 296 (1973). (51) R. L. Harris and J. J. Lagowski, J. Phys. Chem., 82, 729 (1978).
The Journal of Physical Chemistry, Vol. 85, No. 7, 1981 881
ESR of Metal-Ammonia Solutions
Equation 11 in combination with eq 12 predicts that, as the exchange of the electron between the metal cation site and the solvated electron cavity site becomes significant compared to the correlation time due to the diffusion of ammonia, Tl will decrease and become equal to T2. Since the diffusion time, r d , of the nitrogen nuclei in ammonia is known,52the exchange time, rex,can be calculated. The spin-lattice relaxation time, Tl, is affected only by rcwhich according to eq 12 contains both re, and r,+ At that concentration at which Tl becomes one-half of its greatest value, re, must equal T d since r d should not change with metal concentration, in contrast to T ~ The ~ . diffusion time, Td, is 2.21 x lo-" s at -65.0 "C and 1.32 x s at -33.0 0C,52so that at a concentration of about 0.50 M at -33.0 "C and a concentration of about 0.02 M at -65.0 "C the and 2.21 X s, reexchange times are 1.32 X spectively. Since rexis proportional to the quantity q/(TC), where C is the concentration, Tis the temperature, and q is the v i s c ~ s i t y ,we ~ ~can , ~ ~expect to make a reasonable estimate of rexat any concentration for a constant temperature, assuming the viscosity is relatively constant, by using the formula Tex(-33) = 6.6 X 10-l2/c
(13)
for the dilute region at -33.0 "C,and rex(-65) = 4.4 x 10-13/c
(14)
for the dilute region at -65.0 "C; ~ ~ ( is7the 9 exchange time at temperature T. The difference, 1/T2- l/Tl, at both -65.0 and -33.0 "C M, thus, f M must be greater at -65.0 is about lo+ s-l at than at -33.0 "C to compensate for the difference in the exchange times. The equilibrium constant for the overall process e- + M+ + M+e- M. (15) can be described in terms of C, the total metal concentration, and fM, the fraction of electrons at the cation sites. Rearranging the equation for the equilibrium constant, K, gives fM
= KC(1 - fM)2
(16)
which for small f M becomes f M a C. Since l/rexa TC/q, the difference between T1 and T2 should decrease proportionally to the decrease in metal concentration. Eventually TI should equal T2,but the data reported here are apparently still too concentrated to observe this coalescence. Estimation of Electron Solvation Number. The solvation number of the ammoniated electron has been estimated12J5~25~54~55 by using eq 1and 2. Since most estimates either have used T2 or have been obtained by using results from the more concentrated region where T1 = T,,esti(52) J. L. Carolan and T. A. Scott, J. Mag. Reson., 2, 243 (1970). (53)D.Kivelson, J. Chern. Phys., 33, 1094 (1960).
(54)D.E. OReilly, J. Chern. Phys., 41,3736 (1964). (55)R. Catterall, "Metal Ammonia Solutions", Proceedings of Colloque Weyl 11,J. J. Lagowski and M. J. Sienko, Ed., Butterworths, London, 1970,pp 105-130.
mates of the solvation number have been in error by the difference in T1 and Tz in the dilute region; this corresponds to the assumption that eq 11contains only the first term. For wor,
