K. HIGASI,K. BERGMAl\” .4ND
880
no more time consuming than the usual shakeouts (e.g., the cells can be run overnight and sampled in the morning), and in addition they even have Some Positive advantages. For example, they have never given phase disengagement difficulties, they
c. P. SMYTH
Yol. 64
seldom become cloudy, and entrainment troubles are simply non-existent. Acknowledgments.-The authors are indebted to G, N, casefor technical assistance, and to B ~ ~ bara S. Carmack for numerous carefullv Derformed fluorimetric uranium analyses.
JIICROWAVE ABSORPTION ,IXD MOLECULBR STRUCTURE IN LIQUIDS. XXXII. ANALYSIS OF THE REE,lXATIOY TIMES OF n-ALKYL BROMIDES IS TERMS OF A DISTRIBUTIOS BETWEEK LIJIITIXG VALUES’ BY
KENITIHIGASI,I h A U S BERGMASN AND CHARLES P. SMTTH Frick Chemical Laboratory, Princeton University. Princeton, Ai’. J. Received January 11, 1960
The previously measured dielectric constants and losses of the n-alkyl bromides may be represented in terms of a distribution of relaxation times between two limiting values, which may be calculated from the relaxation time and the distribution parameter LY obtained previously from the Cole-Cole arc plot. The lower limit is taken as the relaxation time of the rotational orientation of the CH2Br group about its bond to the rest of the molecule, while the upper limit is the relaxation time of the largest orienting unit, usually the molecule as a whole. The numerical values obtained for the two limits, one small and increasing slowly with molecular size, and the other large and increasing rapidly with molecular size, are consistent with this physical picture of the relaxation process, indicating the approximate correctness of this distribution function.
Dielectric constant, and loss measurements at 1.27, 3.22 and 10.0 cm. wave lengths carried out in tahis L a k ~ m t m r y ~on - ~ eleven n-alkyl bromides were e ~ a m i n e d on ~ . ~t,he basis of the arc plots of Cole and Cole.? The present paper describes a new analysis and interpretation of the same experimental data by the use of a function for the distribution of relaxation times similar to that discussed by Frohlich.* Because of the existence of internal rotation an alkyl halide molecule should have more than one relaxation time and the number of relaxation times should increase with increase in chain length. It is natural to assume that the relaxation times of a straight-chain alkyl bromide are distributed bet,ween two extreme values. The lower limit, rl, is the relaxation time for the rotation of the smallest polar unit, the CH2Br group, situated at the end of the molecule. The upper limit, Q, which is the relaxation time of the largest unit’, corresponds roughly to the end-over-end rot,ation of the whole molecule in the extended form. The probabilities of occurrence of r1 and T~ in the actual relaxation mec*hnnismare not, equal since t8heformer should greatly exceed the latter. Increase in the number of carbon atoms in the molecule increases (1) This research was supported in part b y t h e Kational Science Foundation, in part b y t h e Office of Ordnance Research and in part by t h e United States Air Force through t h e Ofiice of Scientific Research of t h e Ail Renearch and Development Command. Reprodurtion, translation, publication, use or disposal in Ivhole or in part b y or for t h e United Stateb Government is permitted. (2) W. &I. Heston, Jr., E. J. Hennelly and C. P. Smyth, J . 9 m . Chem. Soc., 7 0 , 4093 (1948). (3) H. L. Laquer and C. P. Smyth. ibid.. 70, 1097 (1918). (4) F. H . Branin, Jr., a n d C. P. Smyth, J . Chem. Phga., 20, 1121 (19517).
1.5) E. J. H m n e l l y , W. M. Heston, Jr.. and C. P. Smyth, J . A m . (’hem. Xoc., 7 0 , 4102 (1448). (13) C. P. Smyth, ”Dielectric Behavior and Structure,” NcQrawI l i l l Rook C o . , New York, N. Y . , 1955. pp. 114-127. (7) K. S. Cole and R . H. Cole, J . Chem. Phys.. 9, 341 (1941). ( 8 ) H. Frohlich, “Theory of Dielectrics,” Oxford Univereity Prcrs. London, 1949, pp. 93-95.
the length and variety of the molecular segments which may rotate with the C-Br dipole, but the probability of occurrence of such rotation decreases with increase in the size of the rotating unit, hence with increase in the magnitude of the corresponding relaxation time itself. This picture is in qualitative agreement with a distribution function y ( ~ ) definedgby y(7) =
1 27 if 71 < 7 < rt
y(7) = 0, if r
72
(1)
where A is a parameter which is associated with the distribution of relaxation times. Although the observed dielectric constants and losses of the alkyl halides have been treated by the Cole-Cole method with fair success, their examiriaticn in terms of y ( r ) , which is based on an entirely different mechanism, is revealing. As will be seen, the two very different methods may be equally effective in an empirical representation of the same experimental results. In certain cases we can transform with fair precision t hc two limiting relaxation times, 71 and r 2 , into the relaxation time 7 0 and the distribution parameter cr of the Cole-Cole method and vice z’wsu. 7 0 is the reciprocal of the angular frequency a t nhich the observed dielectric loss has its maximum value e
It
m.
When the values of E” at different frcqiimcies are plotted as ordinates against the corresponding values of e’ as abscissas, the arc of R circle is obtained for a substance which obeys the Cole-Cole relation. In contrast to this, it has becn shown by Bergrnann’O that, for a material obeying eq. 1, when E” is plotted against E ’ the curve obtained is very close to a half-ellipse with the a w s b = (eLl - e r n ) / 2 and u = e”,, if the parameter ,I is not (91 Compare ref. 8
(IO) K. Bergmann, Thesis, Freiburg/Breisq., K r s t Grrniany, 1957.
RELAX~TION TIMESOF +ALKYLBROMIDES
July, 1960
very large, that is, if A
