NOTES
4386 as the phenomenon of spontaneous desorption (Figure 2), present some interest from a thermodynamic viewpoint. The direction of the flux of COz is reversed when the spontaneous desorption begins. The chemical potential of COZ in the solid must then increase at this moment, while the pressure and therefore also the chemical potential of COZ in the gas reach a minimum and begin to increase at the same time. The chemical potential of the zeolite has to decrease simultaneously, since the Helmholtz free energy of the closed system can only decrease at constant temperature and constant volume, indicating that an irreversible transition in the solid occurs. Line a in Figure 1, therefore, represents states of the solid that are metastable relative to the equilibrium states at the same composition ( 8 ) and temperature represented by line b. A transition from a metastable to a stable state at constant composition must involve some spatial rearrangement of the particles in the solid. In other words, if the inverted hysteresis loop were a perfect cycle, the solid after evacuation being in exactly the same state as initially, then it should be possible to run through the same cycle repeatedly a t constant temperature, and a perpetuum mobile of the second kind could be constructed. Since this possibility is excluded (only one cycle in the p-s diagram is possible at constant temperature, which cannot be perfect), the state of the solid initially and after completion of the loop must be different at the same composition. A change in structure, however minor, must have occurred. The transition from the metastable to the stable state a t -80” must be exothermic because it can be reversed at 300”. The metastable state becomes unstable at -80” if a certain critical amount of sorbate is present. Polymorphic transitions have been observed in many silicate systems.112 Different polymorphous forms are generally characterized by different optical or thermal properties; in the systems described here a monotropic transition, which is induced at a certain concentration of guest molecules, changes the sorption properties of the solid. A dependence of the equilibrium lattice positions of cations in zeolites on temperature has been observed by Smith, Bennett, and Flanigen;3 a change of the equilibrium positions of the constituents of the solid upon introduction of a sorbate has been found by Smith4and by Meier and Sh~ernaker.~ (1) W. Eitel, “Silicate Science,” Vol. 111,Academic Press, New York, N. Y., 1965, p 44 ff. (2) F. C. Kracek in “Phase Transformations in Solids,” R. Smoluchowski, J. E. Mayer, and W. A. Weyl, Ed., John Wiley & Sons, New York, N. Y . , 1951, p 257. (3) J . V. Smith, J. M. Bennett, and E. M. Flanigen, Nature, 215, 241 (1967). (4) J. V. Smith, J. Chem. Soc., 3759 (1964). (5) W. M. hleier and D. P. Shoemaker, 2. Kristallogr., 123, 357 (1966).
The Journal of Physical Chemistry
Acknowledgment. The author is indebted to G. H. Kuhl and H. S. Sherry for providing the synthetic zeolite materials.
Analysis of Dielectric Measurements in the Presence of a Small Departure from Debye Behavior In
by Edward H. Grantlb Electromedical Division, The Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 10104 (Received July 2.9, 1.960)
The purpose of this note is to suggest methods by which any given set of dielectric data may best be ex: amined in order to detect small deviations from a single relaxation time. To achieve this, some of the previously published materialzawill be drawn upon but new ideas will be also presented. The emphasis will be on simple analytical techniques not requiring the use of a computer so that a set of results can be quickly tested for departures from the Debye theory. Once Debye behavior is contraindicated, a computer may be necessary to fit the data to the appropriate relaxation function. In general, if a given set of dielectric data will not fit a single relaxation time, the explanation is usually given in terms of a distribution of relaxation times, or a subsidiary overlapping dispersion, or both. The most convenient procedure to adopt for the present purpose is to choose a commonly used distribution function and examine its effect on various well known expressions which are valid when the Debye equations hold. The particular distribution function chosen is unimportant since departure from Debye behavior is the point at issue; for convenience the Cole-Cole3 function (which is probably the one most widely used) will be used. The Cole-Cole equations are normally written in the form
e’ =
E,
+
(eo -
[ [ I + x1--or sin c] 2 E,)
1 + 2 ~ 1 - asin-ffx + xZ(1-a) 2
(1)
(1) (a) This work was supported by NIH Grant HE-01253 NONR551(52); (b) on leave of absence from Queen Eli~abethCollege, London. (2) (a) E. H. Grant, T. J. Buchanan, a n d H . F. Cook, J . Chem. Phys., 25, 156 (1957). (b) H . P. Schwan, 2.Naturforsch., 9a, 35 (1954). (3) K. 8. Cole and R. H. Cole, J , Chem. Phys., 9, 341 (1941).
4387
NOTES where x = armand rmis the most probable relaxation time. Alternatively x = Xm/X = f/fm. The other symbols have their usual significarlce. If a is small and points are considered where the dispersion is appreciable (0.1 < x > 1 or x
