Mukherjee, (Mrs) J.

Roy, T.

English

NOPR

Indian Journal of ChemistryVol. ISA, April 1977, pp. 264-266Absorption In Polar Liquids with the Help of Dispersion RelationT. ROY & (Mrs) J. MUKHERJEEPhysics Department, ]adavpur University, Calcutta 700032Received 3 .11ay 1976; accepted 29 October 1976The working out of ,", i.e. the absorbtive part of the complex dielectric constant for polarliquids, is effected through the use of dispersion relation. A simple analytic form of the com-plex dielectric constant not only displays the Cole-Cole curve but is tractable for Hilbert trans-forms also. The method ~ives a calculated value of ," which is very close to the value of e" forN,N-dimethylamine in cyclohexane determined at 9·313GHz and 25°.THE relaxatio I process in polar organic liquidhas been studied theoretically aid experi-mentally. Colel and Drude- measured the di-electric co.istan ts of polar liquids at various radiofrequencies and provided evidence of anomalousdispersion of dielectric constant in these liquids.The theory was first given by Debye3 and later byFroelich4 and Kauzrnann-. The following empiricalexpression represents the complex dielectric constantas a fuictio.i of the frequency of the oscillatingelectric field,€* (w)=€' (w)~i€"(w)€ -€= €co+1°+-' ... (1)2W't'Tuis gives a semicircular plot in the €', €" plane.Further experimental results of Cole and Cole6,Fr'oelrch", Lane and Saxton", Davidson and Cole", etc.,were analysed by Kauzanann-". Golell and Higasil2.A definite departure from the De bye type relation(1) was proposed by Cole and Co le-". Their muchused empirical relation is represented by Eq. 2.(€o- €co) (2)€*(w) = €co+1+(i<U't')l-!X •..where <J. is an empirical parameter and O~ Cl~ 1.This Cole-O::>le also gives a circular plot in the e',€" plane with the centre below the e' axis.Now, the complex dielectric constant as afunction of th e real frequency must satisfy theKram er-Kro.iig'v type of dispersion relation. Thecondition, that the real and imaginary parts beconnected by the Hilbert transforms-", heavilyrestricts the nature of the functions, i.e. their analy-tic characters. From this point of view. Eq. (1)is acceptable but Eq. (2) presents difficulties as itviolates causality.TheoryIf. instead of relation (2). we assume a generalexpression for €* as given by Eq. 3,U(w)E*(W) = a+V(w)+iW(w) ..• (3)where a is a real constant and U(w), V(w), W(w) arereal polynomials of w, we can surmount the difficulty.264The fact that ,,* must become real constant whenco = 00 and co = 0 makes W = 0 andLt U(w)w-+oo V(w)+iW(w) a real constant ... (4)So far, we have discussed the physical requirementsof Eq. (3) which satisfies the dispersion relation.Or more corr ect ly, the subtracted dispersion relationdepending on the asymptotic behaviour of €*(w).Now we stipulate that U(w). V{w) and W(w) be con-nected by a linear relation (5).CIU(w)+C2VtW)+C3W(<u) = 0 ... (5)where C1. C2. C3 are constants. Tile "*({U) soobtai.red, has the property of depicting exactly theCole-Cole plot in the e' -E" plan e. III order to de-mo.istrate this, we note that because of Eq. (3),(' ) I . "U €' V,,-a ,2€ = V +iW or (€' -a)2+,,"2 Uand,," W€"2+{€'_a)2 - TJFrom Eq. 5,C2€' C3 e"C1+("'_a)2+€"2+(,,'_a)2+,,"2 0i.e. (,,'_a)2+"II2_g€' -h,," = 0 ... (6)which is similar to a Cole-Cole curve.It may be noted that the dispersion-theoreticmethod can vastly improve the accuracy of theexperimental determination of dielectric loss. Infact. the determination of ,," involves a lot of errorsand is much less precise than the determinationof its counterpart €'(w). ,,' can be determined quiteaccurately for various values of w. In fact,nowadays, with the development of microwavetechniques, €' can be obtained for very small toreasonably large values of frequencies. Thus Onehas a fair knowledge of €'(w) from w = 0 to co = 00.This may be used in the dispersion intergal to obtain€"(w) for any particular frequency. This integrationhas to be performed numerically.Results and DiscussionAs an application we have redetermined thevalue of €" at a wavelength of 3 em for N,N-di-ROY & l\IUKHERJEE: ABSORPTION IN POLAR LIQUIDSmethylamine ill cycloliexane at 25°, taking standardprecautions. The value of €"(cu) so obtained is,,-" (9'313 GHz) = 0·0606.The values of €'(cu) for various values of cu havebeen taken from published work. The values of,,-'(cu) being known for some known values of co, acurve of best fit was obtained, which asymptoticallylead to tile constant value and which at co = 0 isthe constant EO' The values of €o' at intermediatevalues of cu were then read off from the interpolatedcurvet". These values were numerically integratedusing relation (7) to obtain the value of €" for thegiven wavelength." co f'" €'(cu')dcu'€ (cu)==P '(')n. -co U) W -(UThis is the subtracted dispersion formula. Weused the relation €"(oo) = 0*. The condition €"(0) = 0 is obtained automatically. For calculatingthe values of €'(cu) for negative values of co, we haveassumed €'(-cu) = E'(CU).The values of E'(CU) for N,N-dimethylamine incyclohexane at 25° for various values of co (ref. 17)are given below:••• (7)w' (GHz) E' w' (GHz) E'0 2·18819·313 2·13 34·86 2·07116·2 2·102 70·01 2·06623·98 2·081 OJ 2·062For the purpose of integration, a change of variableis made to avoid the infinite range. We used therelationa(1 +Z)co = (I-Z)where a = 30, so that Eq. (7) is transformed toEq. (8)." _ 2a(1 +Z) f1 €'(Z')dZ'E (Z) - it P -1 (1+Z')(Z' -Z)€'(Z') is the sort of €'(cu') and €'(cu) normalizedto 0 at co = O.The values of e' (cu) for in termediate values of cowere obtained by parabolic interpolation. Finally,... (8)a computer calculation of the right hand side ofEq. (8) gave the value of e". This computed valueis e" = 0·0607 which agrees well with the experi-mental measured value, €" = 0·0606 as givenearlier.The experirnen tal set up is shown in Fig. 1. Thedielectric loss was calculated by the method ofAltschuler!",In order to illustrate the method we shall calculate€"(cu) which, in the Debye" formula, is€"(cu) = (€0-f",)wt"[I+cu2,,2]-1From Debye expression (10) for €'(cu),'( ) (€o--': )€ cu = €", + l+cuZ-r2The one subtracted dispersion relations is, obtained(Eq. 11).-E'(CU) =!:'!.pf'" [E'(cu':-~'(OO)Jdcuit _'" cu (cu -w)co fco bdcu'= -;;P _ co ~' (w-'---cu-)-(I-+-W-12~,,~2)where b = €o-€ ",.In order to evaluate this integral (11) we tookrecourse to contour integration using Eqs. (9) and(10). The integrals On the contours r+ and T: wereevaluated separately and then subtracted to obtainthe principal value. r,. is the contour boundedby the real axis and the infinite semicircle on theupper half plane and T; is the one bounded by thereal axis and the infinite semicircle in the lowerhalf pl~ne, with indentations. The final integralexpressions arefr.= ? 2iti [- ~ + cu(1 ;:2,,2) - 2(i~3W")]Jt:= ~ 2iti [~+cu(1 ~~2,,2) +2(i:3CU,,)]:.P f:", = -itb,,[1+CU2,,2]-1These expressions are in conformity with theresults.... (9)... (10)... (11)... (12)MOVABLE SHORTINGPLUNGER~~~I_ABSORP"IOH[fCEllTHERMOST~TlC8",.H90~BENDFig. 1 - The experimental set up for measuring dielectric constant------------*E"{OJ) = 0 follows from Eq. (7) since E'{W) is an even function.265INDIAN J. CHEM .• VOL. 15A. APR:L 1977AcknowledgementWe are indebted to Dr A. Roy Chowdhuri forhis interest in the work. The numerical calculationswere performed on an IBM 1130 computer.References1. COLE. A. D .• Ann. Phys .• 57 (1896). 290.2. DRUDE, P .• Z. phys. Chern.• 23 (1897). 267.3. DEBYE, P., Polar molecules (Chemical Catalog. New York),1929.4. FROLICH. H .• Theory oj dielectrics (Oxford UniversityPress. London). 1949.5. KAUZMANN.W.• Rev. mod. Pbys .• 14 (1942). 12.6. COLE.K. S. & COLE.R. H .• J. chem. Phys .• 9 (1941). 341.7. FROLICH. H., Theory oj dielectrics (Oxford UniversityPress. London). 1949.8. LANE. F. A. & SAXTON. F. A.• Proc. R. Soc. London.213A (1952). 400.2669. DAVIDSON.D. W. & COLE. R. H .• J. chem, Phys .• 11)(1951). 1484.10. KAUZMANN. W.. Rev. mod. Phy., 14 (1942), 12.U. COLE. R. H .• J. chem, Phys., 23 (1955). 493.12. HIGASI, K .• Dielectric relaxation 0- molecular structure(Research Institute of Applied Electricity, HokkaidoUniversity. Sapporo. Japan). 1961.13. COLE, K. S. & COLE. R. H .• J. chem. Phys .• 9 (1941).341.14. KRAMERS.H. A.• Estratto Lagli Atti del Congresso Inter-nazionale de Fisici Como. 1927; KRONIG. R. DE, Pbysica,12 (1946). 543.15. TITCHMARSH.E. C., Introduction to the theory of Fourierintegrals (Oxford University Press. London). 1937.16. In practice actually obtained by parabolic interpolation.See computation procedure following equation(8).17. ALTSCHULER,H. M.• Handbook oj microwave measure-ments. Vol. II (Polytechnic Institute of Brooklyn,USA).

Absorption In Polar Liquids with the Help of Dispersion Relation

http://nopr.niscair.res.in/bitstream/123456789/53031/1/IJCA%2015A%284%29%20264-266.pdf

Absorption In Polar Liquids with the Help of Dispersion Relation

Abstract

Similar works

Full text

Available Versions

NOPR