EVALUATION OF DIELECTRIC DATAFOR LIQUIDSAND SOLUTIONS
April, 1959
literature values refer to benzene as solvent; the viscosity difference from xylene is negligible. (ii) .-Pure chlorobenzene was studied over the frequency range 300 to 900 Me. a t 12’. The results are shown in Table 111.
537
10-’2 sec. a t 12’ from the Cole-Cole circular arc. It is to be noted that their value for e’ a t this
temperature is 5.88 f 0.02, whilst the present work obtains 5.84 f 0.03. This indicates the accuracy to which low dielectric constants can be measured. As this work was being completed the thesis of TABLEI11 F. C. De VOS*~ was received. This describes simiPURECHLOROBENZENE AT 12’ FOP DIFFERENT AIR WAVE lar measurements using a coaxial line with a numLENGTHS (XI) ber of fixed filled cells having Teflon windows. Xl(cm.) a’ 6” 10’0 X e”/f This thesis also provides an excellent survey of other 0.275 3.06 5.90 33.32 high frequency dielectric methods. 5.86 .200 2.86 42.94 Grateful thanks are offered to Messrs. Courtauld’s 5.83 .172 49.84 2.86 Educational Trust Fund for a grant which has 5.81 .124 74.73 3.10 made this work possible. The author also wishes 5.81 .093 99.28 3.10 to thank Dr. Manse1 Davies for his advice and constant encouragement during this work. A Taking the mean e “ / f to be (3.0* 0.1) X 10-lo, maintenance grant from the Department of and e m to be na, ie., (1.522)2 = 2.32, then using Scientific and Industrial Research is gratefully sec. equation 4, 7 = (13.6 f 0.5) X acknowledged, and also helpful correspondence Hennelly, Heston and Smyth2aobtained 12.5 X from Professor Bottcher and Dr. F. C. De VOS. (23) E. J. Hennelly, W. M. Heaton and C. P. Smyth, J . Am. Chsm. Sac., TO, 4102 (1948).
(24) F. C. De Vos, Theah, Leiden, 1958.
THE EVALUATION OF DIELECTRIC DATA FOR LIQUIDS AND SOLUTIONS BYGRAHAM WILLIAMS The Edward Davies Chemical Laboratories, The University College of Wales, Aberystwyth, Wales Rscsivsd Bsptsmkr 86, I968
A number of useful modifications and correlations of current methods of evaluating the dielectric parameters for polar
fi%lute uids and solutions are described and illustrated by examples. These include the comparison and correlation of the solution” and “pure li uid” equations for the relaxation time, simple gra hical methods for determining the relaxa-
tion time and its distributionTactor in the general cases, and the calculation of $pole moments via Guggenheim’srelations using Smyth’s dielectric parameters.
The Dilute Solution and Pure Liquid Equations for a Single Relaxation Time.-The general equations for the dielectric constant (E’) and loss factor (e”) of a polar medium possessing a single relaxation time ( 7 ) are e’ = e m
eo +1 +
em
w272
after allowances for the small solvent absorption, is precisely defined by a single relaxation time. Of the various ways of evaluating the parameters of equations 1, three convenient procedures due to Cole may be quoted: (i) the circular arc plot of e” against e’; (ii) the linear plot of e’ against s” X w ; (iii) the linear plot of e‘ against e”/@. All gave 7 = 353 f 6 X 10-l2 sec. and the corresponding value of (eo e,) = 0.99 f 0.01. Comparison of (lb) and (2) gives
-
Here EO is the static dielectric constant of the medium and em is its value a t an angular frequency (0) so high that all orientational polarization has vanished. For a dilute solution of a solute having dipole moment p in a non-polar solvent, Debye,’ twentyfive years ago, made the approximation ’
In this relation €1 is the static dielectric constant of the non-polar solvent and c is the molar concentration. We wish firstly to compare the relaxation times deduced by the use of equations 1 and 2 for the M solution of tri-ntypical case of a 6.54 X butylammonium picrate in m-xylene at 17”. Measurements in these laboratories from 196 kc./sec. to 1700 Mc./sec. show that this system, (1) P. Debye, “Polar Moleculea,” Chemical Catalog Co., New
York. N. Y.,1929, Chapter V.
(EO
- em)
N?r/.LlC
=:
+
K 6750 ___ kT
where K = (€1 2)2. Using the Debye dilute solution equations we obtained p = 11.4 0.1 D: Maryott2 found for the same solute in benzene p = 11.7 D. Accordingly, using the above value of (EO E m ), K =.(EO - e m ) / 5 . 8 9 X lo-’ = 16.8. Equation 2, in which K = (el 2)2, can be arrived a t in various ways.8 In one of them el replaces e‘ the real (and frequency dependent) dielectric constant of the solution; in another (e1 2)2 is substituted as an approximation for (eo 2)(e, 2). The values of these factors are (e1 2)2 = 18.8; (e’ 2)2 = 29.2 to 19.6 over 2)(em 2) the measured frequency range; (eo = 24.3. It is clear that even the best of these factors (18.8) would give a 7-value 20% below that
*
-
+ ++
+
+
+
+
+
(2) A. A. Maryott, J . Rawarch Natl. Bur. Standards, 41, 1 (1948). (8) C. J. F. B(lttcher, “Theory of Electrio Polariaation,” Elrevier PubCo., 1962, p. 374.
GRAHAM WILLIAMS
538
of the correct value from equations 1. Because of the large dipole moment involved, this example serves to emphasize the general conclusion that r values arrived a t via equation 2 or its equivalents can be markedly uncertain even a t quite low concentrations. One general difficulty in using equations 1 for dilute solutions is that (eo - em) will be quite small and the accuracy of r will be markedly dependent upon the value taken for em, in which the uncertainty is accentuated by the difficultiesof measuring e' with sufficient precision a t the highest frequencies. One approximation which can be (and has been) used for em is, of course no2,where no is the refractive index of the solution. The inadequacy of this particular approximation, and a better value for it, is deduced via the form of the Debye equation for the dipole moment of a solute in dilute solution advanced by Guggenheim4
Vol. 63
the following procedure can be used to evaluate rsolely from the loss factor measurements. For wr 1, the same equation l b becomes
-
- (€0
1
whence and the "best" value of B is readily evaluated. Clearly T
= ( A / B ) ' h ; (eo
-
emj
= (ALZ)'/s
*
butylammonium picrate in p-dioxane: we found sec.; B = 4.58 X lo8 set.-'. A = 1.83 X andsolvent, respectively; A = (EO -no2) - (el--nl2); see. and (EO em) and (A/c), is the value of thisratio on extrapolation Accordingly T = 630 x = 0.290. As €0 = 2.53 experimentally, we have to zero concentration, but we shall assume the solution to be sufficiently dilute to use the actual value e m = 2.24. That this should be identical with the value el = 2.24 we measured for our sample of the of theratio. If, for the moment, we assume (el 2) = (nI2 solvent is confirmed from equation 4 as the meas2) for the non-polar solvent, combination of ured value of (no2 - n12)was (1.4248)2- (1.4241)2 = +0.002, ;.e., it is in this instance negligibly small. equations 2 and 3 gives The Relaxation Time and its Distribution Factor. -Cole and ColeBhave provided one of the standard procedures for determining an empirical factor Comparison of this with the general equation l b representing a distribution of relaxation times and Cole7 has described equally convenient means of leads to graphically evaluating the relaxation time when EO - em = A = (EO - no2) - (€1 - me) there is no distribution involved. The following Le., em = no2+ (e1 - nI2) (4) paragraphs show the application of the latter Thus when equations 2 and 3 are simultaneously method to the general case. For a spread of relaxation times, the complex valid, equations 1 and 2 would give coincident r values if el = n12and em = no2. Primarily these dielectric constant is expressed as latter conditions reduce to el = n12, as for ade(5) quately dilute solutions the conditions e m = el and no = nl are likely to be closely approximated. However, in practice el and nI2 can differ significantly even for the conventionally %on-polar" solvents;6 see Table I. For carbon tetrachloride the difference in these factors is nearly 5% and for dioxane it is 10%. This will be the source of discrepant values -if the assumption em = no2 is used.
-
+
Solvent
Benzene p-Xylene Carbon tetrachloride Tetrachloroethylene p-Dioxane
TABLE I 1, oc. 20 20 20 25 20
€1
ntl
2.284 2.270 2.238 2.300 2.209
2.254 2.236 2.132 2.270 1.991
As already emphasized, it frequently happens for solutions that the uncertainty in the e' values leads to very appreciable error in (eo - e m ) . To eliminate this factor for cases where equation l b applies (4) E. A. Guggenheim, Trans. Faraday Soc., 46, 714 (1949). (6) A. Weissberger, et al., "Organic Solvents," Interscience Publishers, Ino., New York, N. Y . , 1955.
I
em)
nl are the refractive indices of the solution
+
I
WT
As an example we may quote the results obtained for a 1.03 X M solution of tetrGnno and
t
nr
and ( 0 7 0 1 " sin - = 2
Y
Thus equations 6 and 7 are e'
= em
+
(EO
-
--; e'
E m ) cy
rc
(€0
- em) 5Y
(6) K. 8;Cole and R. H. Cole, J . .Cham. Phya., 9 , 341 (1941). (7) R. H. Cole, ibid., 28, 493 (1955).
,
I
I ~
I
r
539
EVALUATION OF DIELECTRIC DATA_FOR LIQUIDSAND SOLUTIONS
April, 1959
Rearrmging
Equation 8 may be used conveniently by plotting wn against (eo - e’)/e” for various n, values. The “correct” value of n is quickly found as that giving a straight line whose slope is (sin mr/2),’ ( T ~ and ) ~ whose intercept is (cos n n / 2 ) TO)^. Not only do these factors provide values of TO but their agreement in this respect provides (together with the linearity of the plot) a quantitative check of the 0.0 1.0 2.0 3.0 adequacy of equation 5 in representing the data. A further check, is available by calculation of TO (E+!) a t the different frequencies using the deduced n 7 1 value. Fig. 1.-Anthrone in benzene (20’): 0 , experimental it is points. Full line corresponds t o n = 0.86: TO = 26.3 X From the role of the factor (EO - e’)/“’ clear that the relation 8 will provide best discrimi- sec.; dotted line corresponds to n = 0.86: 70 = 24.8 X nation in the region where the dispersion (or ab- sec. (Smyth, et d 9 ) . sorption) is appreciable. Some alternative versions of (8) may be noted before its use is illustrated. If the departures from a single relaxation time, i.e., n = 1.00, are small, it is convenient to write n = (1 - d). With the approximations cos d?r/2 E 1.00; [sin dn/2 d r / 2 we then have ((
t
Here XO is the “critical” wave length for maximum e”) and X is the wave length corresponding to the e’ and e‘’ values measured a t the angular frequency w. The logarithmic version is (1
- d ) log A 0 - (1 - d ) log x
log
=
-
[(e+)
$1
(9)
Plotting the r.h.s. (neglecting, in the &st ind ) and XO stance, dn/2) against log X leads to (1 values but such log-log plots are notoriously poor methods of detecting departures from algebraic relations. Reverting to (6) and (7) we have
-
(e’
- em) et’
(to
- em)(a/P) - Em)(?/P)
[l
+
(EO
=
or a
(WT0)n
2
nr
(
~
7
““1
cos -
sin~ ) 2
- 1.21 0.0
I
1.0
2.0
I
3.0
~
With the same approximations as above, one finds
Fig. 2.-Fluorenone in benzene (20 ”): 6,experimental points using n = 0.88; a,experimental points using n = 0.87; o , experimental points using n = 0.86.
GRAHAM WILLIAMS
540 (1
- d ) log x - (1 - a) log xo = log
[(-)
-
$1
(10)
This relation 10 is quoted by Grant, Buchanan and Cook* and they have used it to obtain Xo from the log-log plot. However, this is a less generdly useful version than (9) for, as those authors note, the value of e.. has to be assumed, as well as the initial neglect of d?r/2. The use of equation 8 may now be illustrated with the data due to Smyth, et al., who have represented their results for various solutions in the form e'
eo
-
=
El e1
+ a%
4- aof*
e" = a"j2
Here f2 is the mole fraction of the polar solute: e' is the frequency dependent dielectric constant of the solution, and eo the static d.c.8. of the solvent and solutions. Accordingly, equation 8 is tested by plotting wn against ( 0 0 a')/.". Figure 1 shows such plots for anthrone in benzene7 a t 20". Pitt and Smyth have deduced n = 0.86 and T~ = 24.8 X 10-l2 sec. The plot using ~ 0 gives . ~a good ~ straight line whose slope and intercept correspond t o TO = 26.3 X 10-l2 sec.: the line for n = 0.86and TO = 24.8 X 10-l2 sec. is also shown. The slightly higher value of TOis confirmed by individualvcalcuM o n s from equation 8 using n = 0.86: see Table
-
11.
Vol. 63
TABLIE I11 RELAXATION TIMES(IN 10-11, SEC.) FOR FLUORENONE IN BENZENE (20") CALCULATED AT VARIOUSAIR WAVE LENQTHS (A) FOR ASSUMED VALUES OF 12 1.00 0.90 0.88 0.87 0.86 0.84
n X(cm.)
1.25 3.22 10.00 25.00 50.0
0.80
TO
19.9 25.2 33.7 54.1 79.5
20.8 22.9 23.0 28.0 29.8
20.9 22.4 20.4 22.9 20.9
20.9 21.8 19.6 20.4 17.4
20.9 21.4 18.20 18.20 13.2
21.2 20.8 16.1 14.0 6.1
21.2 19.1 12.6 6.3 -ve
For ffuorenone in benzene a t 40" Pitt and Smyth give n = 0.88, TO = 14.9 X sec. The against slope and intercept of the plot of (a0- cy')/a"each give TO = 15.5 X 10-l2 sec. Calculation of Dipole Moments from Smyth's Parameters.-In addition to the relations already quoted for the dielectric properties Smyths has used the relation naD
= ?ZqDi
+ UDfi
for the refractive indices (nD)of solutions in terms of that of the solvent ( n ~ and ~ ) the mole fraction of solute (fi). Smyth, et aZ.,'o then calculate the dipole moments from the Hdverstadt and Kumler relations: we merely wish to indiaate that they can, equally conveniently, be deduced from Guggenheirn's version of the dilute solution equation, see equation 3. In that we now have A
=:
(eo
- n*D) -
(€1
- nD,*)
= (ao
- UD)fa
In the limit of very dilute solutions, f2 = cM1/1OOOp TABLE Ir where M Iis the molecular weight of the solvent and RELAXATION TIMESFOR ANTHRONEIN BENZENE(20") p is its density. Thus, insofar as the Smyth CALCULATED AT VARIOUS AIR WAVE LENGTHS(A) FROM 8 1.25 3.22 10.00 25.00 50.0 26.3 26.9 24.4 25.2 12 EQUATION
(cm.) 101*X T ~ ,sec.
parameters represent the dilute solutions and A/c is a constant, we have A/c = (a0 - aD)(M1/1000p). Some values for benzene solutions calculated in this way (pa) are given in Table IV together with those from Halverstadt and Kumler ( p ~ K). +
The poor value a t 50 cm. emphasiies the uncertainty in (ao- a')/a" (ie., (60 - )'E and 6" TABLE IV are small). COMPARISON OF DIPOLE MOMENTS CALCULATED IN BENZENE For fluorenone in benzene at 20" Pitt and Smyth SOLUTIONS (20") find n = 0.87 and TO = 19.9 X sec. Figure 2 PO, debyea PE+K, debres shows plots of equation 8 for n = 0.88, 0.87, 0.86. I-Nitronaphthalenel0 4.00 4.00 The differentiation over n = 0.87 0.01 is not Anthrone 3.62 3.63 clear. The line drawn for n = 0.87 gave TO = Fluorenone 3.35 3.38 20.9 X 10-l2 sec. from its slope: 20.4 X 10-l2 sec. Phenanthraquinone 5.25 5.36 from its intercept. Accordingly Table 111 repreAs both evaluations are based on the Debye sents the variation in the 70 values calculated a t various wave lengths for different n values. From equation which is of limited significance for soluthis the best values appear to be n = 0.87, TO = tions, the agreement is satisfactory. 20.7 X lo-" sec. in agreement with Pitt and Acknowledgments.-The author wishes to thank Smyth's deductions, although n = 0.88, r0 = 21.5 Dr. Manse1 Davies for his discussion of these points X sec. is almost equally acceptable. and the D.S.I.R. for a Maintenance Award.
*
(8) E . H. Grant, T. 3. Buahanan and H. F. Cook, ib
