Uhf and Microwave Relaxation of LiC104
method does not require a high accuracy of D. However, a higher sensitivity exists in regard to the experimental scatter of S and a consideration of the possible extreme values of S yields an error bound which is in our case increased by a factor of 3. As an error estimate, however, this would be very pessimistic. The magnitude of a0 is supported by which are not presented here Other because of minor variations in the experimental technique.
1221
Acknowledgment. I wish to thank Dr. R. Rosel, Mr. R.D. Seddon, and Mr. D.P. Laurie for computer work and Dr. N. Fenner, Aldermaston, for the mass spectrometric analysis. and Notes (1) R. A. W. Haul, Naturwissenschaffen, 255,41 (1954). ( 2 ) V. Freise, Z.Phys. Chem., 4, 129 (1955). (3) L. Miller, Ber. Bunsenges. Phys. Chem., 75, 206 (1971): Nature (London), 243, 32 (1973).
Ultrahigh Frequency and Microwave Relaxation of Lithium Perchlorate in Tetrahydrofuran Herman Farber and Sergio Petruccl" Departments of Electrical €nsineering/€lectr~physlcsand Chemistry, Polytechnic lnstitute of New York, Brooklyn, New York 1 120 1 (Received May 16, 1974; Revised Manuscript Received January 22, 1975)
The complex permittivity of 0.05 M LE104 in tetrahydrofuran (THF) at 25' in the frequency range 0.3-8.5 GHz has been measured. An admittance bridge was used in the frequency range 0.3-1.5 GHz and a power reflection method in the range 2-8.5 GHz. The real and imaginary parts of the permittivity follow a ColeDavidson distribution of relaxation times with an average relaxation frequency fR = 1.5 GHz and a distribution parameter /3 = 0.8. Including data reported in the literature the relaxation amplitude to - tmof the solute is found to be a linear function of the concentration of ion pairs. The Cole-Davidson distribution parameter, p, is calculated by a modified Glarum theory which postulates that the diffusion-controlled collision between ion-pair dipoles couples with the diffusional rotational relaxation. The calculated value of approximates the experimental distribution parameters including those for LiC104-THF-benzene mixtures.
Introduction Electrolyte solutions have been investigated using dielectric spectrometry by several workers, notably by Hastedl in water and by Davis2 in media of very low dielectric constants. The interpretation of the former systems was given in terms of the solvation of the ions and the subtraction of free rotating solvent molecules by the solvation in the coordination sphere of the ions. The work of Davis2 showed in many instances an onset of a distribution of relaxation times. The interpretation was limited to the qualitative assignment of these complexities to the various dipolar species existing in solution as a result of ionic aggregations. However, for 2:2 electrolytes in water, PotteP succeeded in relating the dielectric spectrum to the relaxation of the diffusional rotation of different solvated ion pairs in addition to the water relaxation. This was based on the Eigen and Tamm theory4 and the measurements5 of ultrasonic spectra for the metal (11) sulfates in water with the hypothesis of the multistep solvent substitution mechanism by the ligand. The association constants of 2:2 electrolytes had been determined6 by classical electrical conductance measurements. T o apply the same sequence to other systems, the electrical conductance and the ultrasonic relaxation times were measured7 for the system LiC104 in T H F between -30 and 25'. Both, ion pairs and triple ions, existed in the concentration range of to 10-1 M , and the for-
mation constants for the two species were determined. U1trasonic data were interpreted as the equilibrium between ion pairs and triple ions. The rate constants and the activation energy indicated that the process was diffusion controlled. The uhf and microwave data reported in this work are used to develop a microscopic interpretation of the dielectric spectra. The current data for 0.05 M LiC104 solutions adds to the available data for 0.25-0.6 M solutions at 30°.8 I t was important to have data for the same system in a broad concentration range, both to check on the reliability of the results and for the interpretation. Experimental Section
Preparation of Materials. Every precaution was taken to ensure the purity of the chemicals and to eliminate traces of water. The tetrahydrofuran (THF, Matheson Coleman) was distilled in a dry nitrogen atmosphere over a K-Na (60% K) alloy in a 3-ft Vigreaux column. Anhydrous reagent LiC104 (Smith, Cleveland, Ohio) was kept for 24 hr a t 60' and 1 Torr. No appreciable loss of weight was noticed with respect to the starting material. Some of the samples of salt were kept at 190' and at atmospheric pressure for 1-2 days with no change. Solutions were prepared by dissolving a dried weighed sample of LiC104 in freshly distilled T H F in volumetric flasks. These operations were The Journal of Physical Chemistry, Vol. 79, No. 12, 1975
Herman Farber and Sergio Petrucci
1222
performed in a dry Nz filled drybox. Contact with the atmosphere was never longer than 30-60 sec when filling the dielectric cell. To check the reproducibility of the data, the dielectric measurements were repeated for two-three different, freshly prepared solutions over a period of 1year. U h f Measurements. The complex permittivities of T H F and of the solutions of LiC104 were measured in the frequency range of 0.3-1.5 GHz using a General Radio uhf admittance bridge, Model 1602B. The details for making this measurement are given by Mopsickg and Glarum.lo T h e Dielectric Cell. A GR874 (D20L) tuning stub was modified by a set of flanges which were used as mounts for a Teflon window. A water jacket also was added to maintain constant cell temperature. The completed dielectric cell and the admittance bridge were mounted on a rigid reference frame and on a sliding platform, respectively. Procedure. The distance between the bridge and the window of the empty dielectric cell was adjusted so that the equivalent electrical position of the dielectric window was a t the bridge center. After filling the cell, the ce!l length was varied and the cell lengths for conductance minima were noted. These values were reproducible to f0.003 in. and f 0 . 2 mmhos the latest as read on the conductance scale of the bridge. The number of minima that can be determined accurately decreases with increasing loss tangents of the liquid diele~tric.~ Calculation of the Complex Dielectric Constant. The normalized admittance of the dielectric filled cell for these conditions is b sinh a - a sin b cosh a - cos b
the dielectric cell was measured as a function of cell length. The analysis was similar to the one described by Cutnell.ll Equipment. A modified reflectometer set up was used to make the measurements in both frequency ranges. The liquid dielectric cells were similar to the ones used for the uhf measurements. Procedure. The reflected power from the dielectric cell was measured continuously as the cell length was varied until a final length was reached when the power no longer varied (Figure 1).The maximum useable cell length is dependent on the loss tangent. For high loss tangents the total microwave power is almost completely absorbed in a short dielectric cell and the measured reflected power comes from the dielectric-window-air interfaces. Data Reduction to Calculate E‘ and E”. The reflected power periodically goes through maxima and minima and the periodicity of each of these is equal to X/2. Cutnellll has shown that the maximum points ( n odd)
or
A semilog plot of (r - Fm)/(l - r,r) vs. n for n odd or F r)/(l r,r) vs. n for n even will have a slope of -a~I/3= -ax12 and an intercept of (aI/3)R.(See Figure 2.) Small deviations of the value of the intercept are the result of a nonideal window and window support, and a nonideal sliding short circuit.ll The same equations for calculating the complex permittivity from a , /3, and in the uhf case also apply in the microwave frequency range.
and for the minimum points ( n even)
where a is the attenuation constant, p is the phase constant = 2 ~ 1 1 R, is the phase change of the reflected wave = T for an ideal short circuit, r is the voltage reflection coefficient, and I‘, is the voltage reflection coefficient for a dielectric cell which is electrically infinitely long. The plus sign is for n even. chosen when l7, < e-(a/fi)(n?r-R) A convenient reference power level is the reflected power when the electrical length of the dielectric cell is infinite
w-1.
(r,
where Go, the characteristic transmission line conductance, is equal to 20 mmhos; G is the conductance as measured on the bridge; a = 2al; b = 2/31; 1 is the length of the dielectric cell; XO = free space wavelength; X = wavelength in dielectric medium; and y = O( j p , the complex propagation constant of the dielectric filled line. The values of G, are the bridge readings at the bridge balances (absolute minima), 1 = (nX/4) where n = 1, 3, 5
+
....
The least-squares plot of 1 vs. n gives as a slope dlldn = Xl4. The function, Cn(a,b), has been tabulated by Mopsikg for n = 1 , 3 , and 5 . A computer program has been set up for larger values of n up to n = 11 giving the values of a2 - b2 and 2ab. The real and imaginary parts of the complex dielectric j c ” may then be calculated from the folconstant E = E’ lowing equations:
+
These calculations may be simplified as follows: for low loss dielectrics C,(a,b) = nra12. For the conditions a > KT > Kquadrupole.27 L is the Avogadro number; the factor 1h accounts for the fact that the ion pair is considered alternatively cavity and rotating dipole. The values of 10 and of TD for c = 0.05 M at 25' measured in this study and at c = 0.25-0.40 and 0.60 M at 30' are reported in Table 111 together with the calculated values of /3 from eq I. The calculated and the experimental values of p are in agreement with the exception of the value of c = 0.4 M a where the experimental results show also an inversion in the trend of 6 - with concentration. Also as the concentration increases 7dif becomes shorter and closer to the value of TD. The process becomes cooperative and tends toward 0.5 as predicted by the Glarum theory.25 In fact, recently Lestrade et a1.28have published data for LiC104 in benzene added to T H F as to make the molar ratio (THF/LiC104) = 3. The electrolyte concentration was c = 0.8 M . By setting t m for the LiC104 solution equal to the pure solvent ( 6 - = 2.38) these authors give T D = 5.10 X sec, (3 = 0.51. Setting c m = 2.274, the permittivity of pure benzene, it gave TD = 5.20 X sec and p = 0.50. Using the modified Glarum theory we obtain lo = 8.04 X cm and 7dif = 4.97 X sec, which in turn gives for 7~ = 5.1 x IO-1o sec &,lcd 0.50 and for 7D = 5.2 x sec, (3 = 0.49. References a n d Notes (1) J. B. Hasted and G. W. Roderick, J. Chem. Phys., 29, 17 (1958)
Pexpt
0.86 0.74 0.67 0.65
0.80 0.79 0.53 0.65
Ref b
f f i
0.06 0.04
C
0.05
C
C
(2) M. Davis and G. Johansson, Acta Chem. Scand., 18, 1171 (1964), and previous literature quoted therein. (3) R. Pottel, "Chemical Physics of Ionic Sdutions",B. E. Conway and R. G. Barradas, Ed., Wiley, New York, N.Y., 1966, p 581. (4) M. Eigen and K. Tamm, 2.Electrochem., 66, 93 (1962). (5) K. Tamm, G. Kurtze, and R. Kaiser, Acustica, 4, 380 (1954); G. Kurtze and K. Tamm. ibid.. 3. 33 11954). (6) G.Dav,;i "Ion Association", Butterworths, London, 1962, pp 168-172. (7) P. Jagodzinski and S. Petrucci, J. Phys. Chem., 78, 917 (1974). (8) J. P. Badiali, H. Cachet, and J. C. Lestrade, C. R. Acad. Sci., Paris, 271, 705 (1970); J. P. Badiali, H. Cachet, and J. C. Lestrade, Ber. Bunsenges. Phys. Chem., 75, 297 (1971). (9) F. I. Mopsik, Ph.D. Thesis, Brown University, 1964; F. I. Mopsik and R. H. Cole, J. Chem. Phys., 44, 1015 (1966). (IO) S. H. Glarum, Rev. Sci. lnstrum., 29, 1016 (1958). (11) J. D. Cutnell, Ph.D. Thesis, University of Wisconsin, 1967; J. D. Cutnell, D. E. Kranbuehl, E. M. Turner, and W. E. Vaughan, Rev. Sci. Instrum., 40, 908 (1969). (12) D. J. Metz and A. Glines, J. Phys. Chem., 71, 1158 (1967); E. Kuss, Z. Angew. Phys., 7, 376 (1955). (13) J. P. Badiali, H. Cachet, A. Cyrot, and J. C. Lestrade, J. Chem. SOC., Faraday Trans. 2,69, 1339 (1973). (14) E. A. S. Cavell, Trans. Faraday Soc., 61, 1578 (1965); E. A. S. Cavell. P. C. Knight, and M. A. Sheikh, ibid., 67, 584 (1971). (15) N. Hill, W. E. Vaughan, A. H. Price, and M. Davies, "Dielectric Properties and Molecular Behaviour", Van Nostrand, London, 1969, pp 55, 244. (16) E. A. S. Cavell and P. C. Knight, J. Chem. Soc., Faraday Trans. 2, 68, 765 (1972). (17) R. Kubo, J. Phys. SOC., 12, 570 (1957): "Lectures in Theoretical Physics", Vol. I, Interscience, New York, N.Y., 1958, Chapter IV. (18) J. Barthel, H. Behret, and F. Schmithals. Ber. Bunsenges. Phys. Chem., 75, 305 (1971). (19) . . E. A. S. Cavell. P. C. Kniaht, and M. A. Sheik, Trans. Faraday SOC., 67, 2225 (197 1). (20) C. A. Kraus, et. al., J. Am. Chem. SOC.,72, 166 (1950); 55, 21, 3614 (1933). (21) M. von Smoluchowski, 2. Phys. Chem. (Frankfut? am Main), 92, 179 (1917); P. Debye, Trans. Elect. SOC.,82, 265 (1942). (22) R. M. Fuoss, J. Am. Chem. Soc., 80, 5059 (1958). (23) The present data of molar refraction taken at the sodium D line wavelength are RD(Li+) = 0.20 cm3, Ro(C104-) = 13.32 cm3 (from C. P. Smyth "Dielectric Constant and Molecular Structure", Chemical Catalog, New York, N.Y., 1931). (24) G. Schwartz. J. Phys. Chem., 71, 4021 (1967); 74, 654 (1970); G. Williams, Adv. Mol. Relaxation Processes, l,409 (1970); W. Scheider, J. Phys. Chem., 74,4296 (1970). (25) S. H. Glarum, J. Chem. Phys., 33, 639 (1960). (26) The rotational relaxation time is proportional to the square root of the moment of inertia [E. Bauer, Can. Phys., 20, 1 (1944)]. (27) H. C. Wang and P. Hemmes, J. Am. Chem. SOC.,95,5119 (1973). (28) J. P. Badiali, H. Cachet, P. Canard, A. Cyrot, and J. C. Lestrade, C. R. Acad. Sci., Paris, 273, 199 (1971).
The Journal of Physical Chemistry, Vol. 79, No. 12, 1975
