1496
NAOKAZU KOIZUMI AND TETSUYA HANAI
Vol. 60
DIELECTRIC PROPERTIES OF LOWER-MEMBERED POLYETHYLENE GLYCOLS AT LOW FREQUENCIES BYNAOKAZU KOIZUMI AND TETSUYA HANAI Institute for Chemical Research, Kyoto University, Takatsuki, Osaka-Fu, Japan Received April $4, 1966
The dielectric roperties of seven polymeric homologs of glycol from ethylene glycol to heptaethylene glycol have been measured at the frequencies of 0.5 to 50 kc./sec. by means of a conductance ca acitance bridge. The measurements were made on liquid, supercooled liquid and solid states of these polyethylene glycoE over the temperature range from -70 to $60”. I n the liquid state the dielectric constant for each polyethylene glycol showed the equilibrium or static value to be dependent on temperature, being smaller, the higher the homolog of polyethylene glycol. It was found empirically that the variation of static dielectric constant for these polyethylene glycols was linear with the reciprocal absolute temperature. I n the crystalline solid the dielectric constant showed small values of 3 to 4 almost independent of the type of polyethylene glycol, and a small dielectric dispersion was observed near -60’ for glycols higher than tetraethylene glycol. I n the case of tetraethylene glycol the dielectric dispersion and absorption which can be expressed by the “lemniscate” formula was observed in the superrooled liquid state.
Introduction Although some extensive investigations have hitherto been carried out on the dielectric properdielectric measurements ties of high polymers, of lower polymeric homologs have not been made. It may be important for studying the dielectric properties of high polymers to obtain some information about the dielectric behavior of lower polymeric homologs. I n this research some dielectric measurements have been attempted on lower polymeric homologs of polyethylene glycol. Polyethylene glycols HO[CH~CHZO],-ICH,CH~OHare characterized by a very hydrophilic or*water-soluble nature and have relatively large dielectric constants. This hydrophilic property may arise from the two end hydroxyl groups and the polyoxyethylene group. Some physical properties of polyethylene glycols have been reported by Hibbert and his co-workers, and others.2 However, there are no available data on dielectric properties of polyethylene glycols. The present work was undertaken to determine the static dielectric constants of lower homologs of polyethylene glycol and examine the variation of their dielectric properties with the series of homoI ogs. Experimental Materials.-The seven homologous polyethylene glycols from ethylene gl col to heptaethylene glycol were prepared b the fractionaf distillation of commercial products from d r b i d e and Carbon Chem. Co. Since possible impurities present in the commercial glycols used are water and homologous glycols, and since the glycols used herein form no aaeotropic mixture with water, it is relatively easy to obtain each homologous polyethylene glycol by fractional distillation of commercial products dried over a suitable drying agent. Commercial ethylene glycol was purified by drying over freshly dehydrated sodium sulfate and three successive fractional distillations a t a reduced pressure, retaining only the middle fraction, and finally recrystallized twice; b.p. 83.0’ (8 mm.); m.p. -12’; lit. m.p. -13.’,2 -13.203; density dz04 1.1150; lit. d% 1.1136.a Commercial diethylene (1) See (a) R. E. Burk and 0.Grummitt, “The Chemistry of Large Molecules,” Interscience Pub. Ino., New York, N. Y., 1943. Chap. VI: (b) F. H. Mllller and Chr. Schmelier, Eroeb. ezakt. Natu?., 26, 359 (1951); ( 0 ) F. WOrstlin, Kolloid-2.. 134, 135 (1953): (d) A. R. von Hippel, “Dielectric Materials and Applications,” John Wiley and Sons. Inc., New York, N. Y., 1954, Chap. V. (2) Cited in G. 0. Curme, Jr., and F. Johnston, “Glyool,” Reinhold Publ. Corp., New York, N. Y., 1952, Chap. 1, 3 and 7. (3) A. F. Gallaugher and H. Hibbert, J . Am. Chem. Xoc,, SE, 813 (1936).
glycol and triethylene glycol were purified in the same way as ethylene glycol. Diethylene glycol: b.p. 117.2-117.5’; (7 mm.); m.p. -6’; lit. m.p. -8’,2 -10.1°.3 Triethylene glycol: b.p. 144 f 1’ (6 mm.); m.p. -8’; lit. m.p. -4.3°,4 -7.2’,2 -9.403; hydroxyl value 748 (theoretical 747.5). Tetraethylene glycol and higher homologs were prepared by several successive fractional distillations of “Polyethylene Glycol 300” and “Polyethylene Glycol 400” from Carbide and Carbon Chem. Co. under a reduced pressure of nitrogen, to avoid oxidation by air a t high boiling temperature. Finally, each homolog was purified by two or more fractional recrystallizations, retaining only the crystalline portion. Tetraethylene glycol: b.p. 154 f 1’ (2 mm.); m.p. -3’; lit. m.p. -6.2’,2 -9.4’3; hydroxyl value 581, (theoretical 578.0). Pentaetbylene glycol: b.p. 184 1 (2 mm.); m.p. range -2 to 0 ; lit. m.p. -8.703; hydroxyl value 472 (theoretical 471.1). Hexaethylene glycol: b.p. 214 f 1’ (2 mm.); m.p. range 5 to 7’; lit. m.p. 2.1°,2 1.303; hydroxyl value 378 (theoretical 397.6). Heptaethylene glycol: b.p. 237-244’ (2 mm.); m.p. range 10 to 12’. lit. m.p. 7.7’3; hydroxyl value 330 (theoretical 34379). These glycols are very hygroscopic, so that refined samples were always kept in a desiccator just before use to prevent them from absorbing moisture. Apparatus and Measurements.-Capacitance and conductance measurements were made on the liquid and solid states of glycols a t temperatures from -70 to +60’ over the frequency range of 0.5-50 kc. with an impedance bridge of the parallel resistance type which has been described elsewhere.6 The measuring cell used consisted of two concentric platinum cylinders whose empty capacitance was about 30 ppf., and was calibrated with several standard liquids.6 The accuracy of the dielectric constant was within f 0 . 5 % . The bridge used was rather suitable for measurements of high conductance which ranged from 0.1 to 1000 pmhos. To check the value of equivalent high frequency conductance in measurements of dielectric absorotion, the equivalent capacitance Cp and conductance Gp for parallel combination were measured on a series combination of resistance R. and capacitance C, as a function of frequency. I t is easily shown that C , and Gp are related to C. and R, by the equatidns
where w is the angular frequency equal to 27r times frequency T. is the time constant for given C. and R.. Therefore, plots of G,/w versus C, as a function of frequency must fall on a semi-circle for given C, and.%, and plots of log (GD/uCP)versus log f must give a straight line with a slope of unity, whose intercept with the line for log (GP/uCp) = 0 gives the frequency corresponding to the reciprocal of the time constant. The test plots of GP/w
f and
(4) Carbide and Carbon Chem. Co., “Physical Properties of Synthetic Org. Chemicals,” 1956 ed. (5) N. Koizumi and T. Hanai, Bull. Inel. Chew. Research, Kyoto Univ., S3, 14 (1955).
Nov., 19SG
DIELECTRIC PROPERTIES OF LOWER-MEMBERED POLYETHYLENE GLYCOLS
1497
TABLE I DIELECTRIC CONSTANTS OF HOMOLOGOUS POLYETHYLENE GLYCOLS Temp., %.
60 50 35 20 5 - 5 -20
31.58 (lit.6) 33.21 (lit.6) 35.86' (lit.6) 38.66 (lit.8)
Diethylene
33.42 35.28 38.41 41.82 45.70 48.50 53.01
Triethylene
e
of glycols
Tetraethylene Pentaethylene Hexaethylene
Liquid and supercooled liquids" 24.83 19.18 16.53 26.40 20.11 17.26 28.90 21.94 18.84 20.44 31.69 23.69 25.60 22.24 34.81 27.00 23.55 37.12 40.69 29.36 25.51
Solidb 3.93 4.45 3.73 3.47 3.60 3.52 3.32 3.43 3.43 3.24 3.39 3.37 3.21 3.41 3.33 3.21 3.37 3.31 3.21 3.24 Static dielectric constants. b Values at 5 kc. 0 Interpolated value.
- 10 -20 -30 -40 -50 -60 -70
a
Dielectric constant -Ethylene-
...
versus C , and log (G,/wC,) versus log f obtained for a series
combination of a carbon resistor and a mica condenser were consistent precisely with the predicted forms. It was, of course, checked beforehand that the resistor and the capacitor alone had no frequency de endent value. This procedure will be useful to check whether the measuring set gives systematic errors in measurements of the equivalent high frequency conductance for dielectric absorption or not.
Results and Discussion Liquid.-The dielectric constants observed for polyethylene glycols at 5 kc. are listed in Table I. These glycols showed a sharp change in dielectric constant at the melting point, giving small values in the solid state which were almost independent of kinds of glycol. I n the liquid and supercooled liquid states above temperatures of about -30", the dielectric constant showed no variation with frequency, within the present frequency range, corresponding to the static dielectric constants, and showed the normal decreasing tendency with increasing temperature which is characteristic of dipolar liquids. The higher the homolog of polyethylene glycol, the smaller was the dielectric constant. Considering that the main contribution to the dielectric polarization of liquid polyethylene glycol may be due to the orientation polarization of hydroxyl groups, it is a reasonable result that higher polyethylene glycols, having fewer hydroxyl groups per cc., show smaller dielectric constants. There are no available data on the dielgctric constants of polyethylene glycols. Although Akerlof's data0 for ethylene glycol are accepted as the static dielectric constants, the values observed in this work were somewhat larger than those of Akerlof, as given in Table I. For associated !iquids such as polyethylene glycols which involve strong intermolecular hydrogen bonding, the relation between the dielectric constant and the dipole moment may be considered (6) G . Akerlbf, J . A m . Chem. Soc., 64, 4130 (1932).
3.80 3.49 3.36 3.34 3.36 3.27 3.14
Heptaethylene
14.83 15.55 16.86 18.16 19.60 20.67 22.30
13.28 13.86 14.92 16.00 17.14 18.00 . 19.48
12.45 13.01 13.96 14.85 15.83 16.66 17.90
4.61 4.21 3.99 3.82 3.66 3.31 3.24
3.96 3.91 3.87 3.83 3.73 3.39 3.27
3.96 3.91 3.88 3.85 3.79 3.56 3.41
quantitatively in terms of the Kirkwood relationship' ..where M is the molecular weight, d the density, N Avogadro's number, a the optical polarizability, p the dipole moment in liquid state and g the correlation parameter. The calculated value of the effective dipole moment in liquid g'/gp for each polyethylene glycol increased with the increasing number of the polymeric member, as shown in Table 11. TABLE 11 LIQUIDMOMENTS OF POLYETHYLENE GLYCOLS BY WOOD'S RELATIONSHIP
KIRK-
Glycol
Liquid moment g1'*p X 10'8. e.8.u. e t 20°
Ethylene Diethylene Triethylene Tetraethylene Pentaethylene Hexaethylene Heptaethylene
4.87 5.50 5.58 5.84 6.05 6.14 6.35
Uchida, Kurita, Koizumi and Kubo8 have recently measured the dipole moment of the same specimens used herein in dioxane solution and discussed the relation between the dipole moment and the molecular structure. The results showed the same tendency of the dipole moment to increase. It was found that the observed dielectric constants for all the polyethylene glycols used were very linear with the reciprocal of absolute temperature T i n a temperature range from -30 to $60'. The numerical values of the dielectric constant for the individual polyethylene glycols are given as a function of 1/T by the formulas. (7) J. G . Kirkwood, J . Chem. Phys., 7 , 911 (1939); Trans. Faraday 42A, 7 (1946). ( 8 ) T . Uchida, Y. Kurita, N. Koiaumi and M. ICubo, J . Polumsr Sci., in press. Soc.,
NAOKAZU KOIZUMI AND TETSUYA HANAI
1498 Ethylene glycol E Diethylene gl col E Triethylene gLcol E Tetraethylene glycol e Pentaethylene glycol E Hexaethylene glycol E Heptaethylene glycol E
20700 X 16797 X 10912 X 9731 X 8074 X 6552 X 5700 X
1/T
Vol. 60
- 28.75 - 13.57
like. Accordingly, in the case of diethylene glycol one of the two end hydroxyl groups would take 1/T part in d.c. conduction effectively. The thermal = 1/T - 12.75 evidence for the intramolecular hydrogen bonding = 1/T - 9.40 = 1/T - 6.38 of diethylene glycol already has been reported by = 1/T - 4.62 Hibbert and his co-workers. la The maximum difference between the observed It appears, of course, that intramolecular hydrovalues and those calculated by the above formula gen bonding is also possible in the case of higher was within -10.5%, being of the order of the ex- polyethylene glycols. perimental error in this work. As for the temperature dependence of the conThe linear relation of dielectric constant to 1/T ductivity for polyethylene glycol, plots of log u verwas presented by Morgan and L o w r ~ and , ~ Cole, sus 1/T showed some deviation from linearity. et aZ.'O; no theoretical basis was given. The Discussion of this point is given in the last section of present authors have also found that this linear re- the paper. lation holds for many other dipolar liquids. YaSolid.-On cooling polyethylene glycols through sumi and Komooka" recently have developed a the melting. Doint. the liauid state continued in vartheory on the dielectric constant of polar liquids ious tempgrature 'range; and the dielectric constant which gives the linear relation of dielectric constant rose gradually, but a t a temperature of about to 1/T. -30" below the melting point the liquid solidified The conductivity for polyethylene glycols meas- into a crystalline state, giving a small value of 3 to ured together with the dielectric constant in the 4 as shown in Table I. liquid state was also independent of frequency This indicates that in the crystalline solid of within the present frequency range, corresponding polyethylene glycol most of the molecular dipoles to d.c. conductivity. I n many cases it is often are rotationally frozen or have no freedom of oriendifficult to make absolute measurements of the tation to the electric field. conductivity of a pure organic liquid, since any It was often noted that diethylene, triethylene trace of ionic impurity affects the conductivity and tetraethylene glycols supercooled until about considerably. I n the present work the numerical -50°, setting to a transparent vitreous state. value of conductivity was reproducible within 10% This vitreous state continued for several hours when no external mechanical shock was given, and thereirrespective of the purification run. The conductivity for homologous polyethylene after a white portion of spherical form, that is, a nuglycols decreased with higher members of the homol- cleus of polycrystal, appeared in the vitreous solid. ogous series except in the case of diethylene gly- But the nucleus grew too slowly, and a few days would have been necessary for the whole system t o col, as given in Table 111. crystallize a t about - 50". On warmingup to about TABLE I11 20°, however, the nucleus of crystal grew very rapD. C. CONDUCTIVITIFS OF HOMOLOGOUS POLYETHYLENEidly. Such behavior of crystallization also has been GLYCOLS AT 20" found for isobutyl chloride14and glycerol.l6 It was D.c. oonductivity X 108, considered that due to the very high viscosity of the mho om.-' Glycol This research Lit.18 vitreous state the rearrangement of molecules Ethylene 16 116 (25") hardly occurred a t a low temperature, but a t a Diethylene 3.1 higher temperature below the melting point the Triethylene 8.4 decrease in viscosity made the molecular rearTetraethylene 4.7 rangement, and thus the crystal growth, relatively Pentaethylene 3.0 easy. Hexaethylene 1.3 In dielectric measurements of the solid state, it is Heptaethylene 1.o important t o attain thermal equilibrium. For this reason, when the temperature of the surroundIt appears likely that most of the d.c. conductiv- ing bath around the measuring cell was changed and ity for glycols is attributable to the proton conduc- set to a desired value, the measurements were made tion mechanism by the intermolecular hydrogen a t a time interval of ten minutes or more until no bonding of the hydroxyl group. Therefore, for variation of dielectric constant was observed with higher polyethylene glycols the number of hydroxyl the lapse of time. Particularly a t the temperature groups per unit volume becomes smaller, resulting just below the melting point it was often noted that in the lower d.c. conduction. the temperature equilibrium was attained after ten Only diethylene glycol showed relatively small hours or more. The values as shown in Table I conductivity, being of the order of that for penta- are those obtained in this way. ethylene glycol. This small d.c. conductivity the lower glycols such as ethylene, diethylmight be caused by intramolecular hydrogen bond- eneFor and triethylene glycols the dielectric constant ing between two end hydroxyl groups, which was about 3.3 in the crystalline solid state, dewould make the diethylene glycol molecule ring- creasing only slightly with decreasing temperature (9) s. 0. Morgan and H. H. Lowry, THISJOURNAL, 34,2385 (1930). within a temperature range from the melting point (10) F. X. Hassion and R. H. Cole, J . Cham. P h w , 23, 1756 (1955); to -70°, and no dielectric loss was observed. D. J. Denney and R . H. Cole, ibtd., 23, 1767 (1955). = = =
1/T - 25.59
*
(I
(11) M. Yasumi and H. Komooka, BUZZ. Cham. SOC.Japan,29, 407 (1956). (12) R. M. Mtiller, V.Rsschka and M. Wittrnann, Monatsh. CApm., 48, 659 L1927).
(13) A. F. Gallaugher and H. Hibbert, J . A m . Chem. Soc., 69, 2514, 2521 (1937). (14) A. Turkevioh and C. P. Smyth, ibid., 64, 737 (1942). (15) A. K. Schulz, Z. Nalurforschg., 90, 944 (1954).
I.
DIELECTRIC PROPERTIES OF LOWER-MEMBERED POLYETHYLENE GLYCOLS
Nov., 1956
5
0
10
20
15
25
30
1499
35
d. Fig. 1.-Plots
'
of
e" versus E' for supercooled tetraethylene glycol. Numbers beside plots denote frequency in kc. Solid curve was drawn by eq. 1 for the case: EO = 33.7 (at -56'), em = 4.0 and ~ p / 2= 35".
On the other hand, in the case of those higher polyethylene glycols such as penta-, hexa- and heptaethylene glycol, the dielectric constant showed a value of about 3.8 near -40", decreasing at lower temperatures, and becoming about 3.3 at -70". It was considered that the drop in the dielectric constant near -60" for higher polyethylene glycols was due to a small dielectric dispersion, because the high frequency conductance increased appreciably, associated with the drop in the dielectric constant. The maximum dielectric loss was estimated to be of the order of 0.1 a t 5 kc., too small to be measured with accuracy in the bridge used. Therefore the loss data are not given here. Although no conclusive interpretations can be made, there are two possible explanations for the small dielectric dispersion observed for the higher polyethylene glycols. First, it can be considered that freedom of orientation of the polarization due to twisting of the molecular chain around the long axis is possible only for higher polyethylene glycols. Another possibility is that a small dielectric dispersion is caused by the dipole groups of amorphous solid present in the crystalline solid. The more gradual change in the dielectric constant a t the melting_ point shows that a distribution of molecular _ size occurs. A similar small dispersion of the dielectric constant in the solid state has been reported for some other solids, long chain esters,16 ether," amorphous polymers'8 and partially crystalline polyesters. l9 Dielectric Relaxation.-As mentioned in the previous subsection, polyethylene glycols supercooled easily in some temperature ranges and showed a very high viscosity with decreasing temperature. I n such a viscous or vitreous state dielectric dispersion and absorption would be
expected for polyethylene glycols a t the present experimental frequencies. I n fact, the dielectric dispersion was appreciably noted for these glycols below -40" a t frequencies of about 50 kc. However, the supercooled state was not always stable during the dispersion measurements, and crystallization occurred spontaneously. I n some cases with triethylene and tetraethylene glycols a stable supercooled state was realized for a time long enough to measure the complex dielectric constant t* as a function of frequency. The observed values of the complex dielectric constant for tetraethylene glycol are plotted in a complex plane in Fig. 1. The loci of e*( = E' - jd') for tetraethylene glycol fitted approximately to the "lemniscate" formula which has been given for supercooled propylene glycol and glycerol by Davidson and Cole.20s21
(16) R. W. Croive and C. P. Srnyth, J . Am. Chem. Soc., 72, 5281 (1950). (17) A. Sohallamaoh, Nature, 168, 619 (1946). (18) E.&, ref. la, p. 450. (19) F. WILrstlin, Koiloid-Z., 110,71 (1948).
(20) D. W. Davidson and R. H. Cole, J . Cham. Phgs., 19, 1484 (1951). (21) 8. Takahashi suggested that eq. 1 for 6 = 0.5 oorresponds to Bernoulli's lerniscate: Research Electrotech. Lab. Japan, No. 539, p. 60 (1953).
=
e* --e,
€0
(1
- Em
+
jW7)8'
l,P>O
where EO is the static value, ern the limiting value at the high frequency side, r the most probable relaxation time, p a parameter for the distribution of relaxation times and j and w have the usual signifid cance. From eq. 1 the real and imaginary parts of the dielectric constant, t' and e", are given as e'
- em e"
- Em)(COS4 ) cos pq5 - em )(cos 4)B sin @+.
= (eo
= (a
'
where tan #I = U T . Then, putting 0 = tan-l[e"/(e' - em)], one obtains another expression for the frequency dependence of E* in the dielectric relaxation of "lemniscate" type W T = tan (e/,@ (2)
NAOKAZU KOIZUMI AND TETSUYA HANAI
1500
VOl. 60
T is to be linear with the reciprocal absolute temperature 1/T; however, some systematic deviation was found for the case of tetraethylene glycol, as shown in Fig. 2. If log T was plotted against 1/(T - T,) where T m is a parameter, it was found that the variation of log 7 was approximately linear with 1/(T T,) (Fig. 2), as is the case for propylene glycol and glycer01.~0 Therefore, the temperature dependence of 7 was given by the expression T = A exp[B/(T - T m ) ] (3) where A and B are parameters whose numerical values are given in Table IV. This situation would show literally that T becomes infinitely large at the temperature Tm or that T, is a characteristic temperature where the orientation polarization disappears owing to the infinitely slow relaxation process. It is also a noticeable feature that a similar relation to that in the relaxation process was found for the temperature dependence of conI . I 1 110-11 ductivity Q of tetraethylene glycol; the plots 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 of log u versus l / T for polyethylene glycols 1/T x 108 (OK.). were slightly convex upwards, but the plot of Fig. 2.-Logarithm of relaxation time T versus 1/T and log u versus 1/(T - T,) for tetraethylene gly1/(T - T,) (T, = -108') for supercooled tetraethylene col was a straight line, within the experimental glycol: 0,values for supercooled state observed in this error. Hence, the expression of the form of work; a,values obtained from microwave d a h z 4 eq. 3 is also valid for the temperature depenAccordingly, log-log plots of tan(s/p) versus w dence-of conductivity. must give a straight line with a slope of unity. TABLEIV Although the lemniscate formula fitted to the complex locus of dielectric relaxation for tetraethyl- PARAMETERS OF TEMPERATURE DEPENDENCE OF RELAXAene glycol more precisely than the ordinary circular TSON AND CONDUCTION PROCESSES FOR TETRAETHYLENE arc rule,22strictly speaking, a slight deviation was GLYCOL noted : The solid curve in Fig. 1 which is calculated Temp. range B Tm (0C.I A (OK.) (OK.) for the case that EO = 33.7 at -56", ern = 4.0 and Measurement r p / 2 = 35O, fitted t o the dispersion data for -56' -62 $50 1.028 X 10-15 994 165 Dielectric at the low frequency side, but a t higher frequencies Conductivity -20 * 4-60 2.96 X 10-6 -829 165 rather fitted to those for -62". Checking the frequency dependence of E* by eq. 2, log-log plots Considering that the d.c. conductivity of polyof tan(6IP) versus frequency showed slightly smaller ethylene glycol is caused mainly by the proton slopes than unity. conduction mechanism which requires the rearIt seems that the exact expression for dielectric rangement of the hydroxyl group, it might be plausrelaxation of supercooled tetraethylene glycol would ible that Tmis -108" for both relaxation and coninvolve a more complicated distribution function of duction processes !Table IV) and both the dipole relaxation times than those derived from lemniscate orientation and the proton conduction hardly ocformula and circular arc rule. The fact that a cur at the temperature of T m . parameter for tetraethylene glycol was smaller, It has been reported also by Davidson and Colez0 about 0.4, than those of 0.66 and 0.6 for propylene and Schulz" that eq. 3 is applicable for the viscous glycol and glycerol, respectively,20 indicates the flow or the rheological process in supercooled glyexistence of a more broad distribution of relaxation cerol. times for tetraethylene glycol. This feature is Acknowledgments.-The authors wish to aclikely related t o the non-rigid molecular chain of tet- knowledge their indebtedness to Professor Remraethylene glycol consisting of a fairly flexible poly- pei Goto for many valuable discussions, and oxyethylene group -[CH2CHt0],-.* also to Mr. N. Hayama for his aid in purification The most probable relaxation time T was obtained of materials. This research has been supported by from the frequency for log tan(6/p) equal to zero. the Scientific Research Encouragement Grant from According to the rate theory of Eyring,2alogarithm of the Ministry of Education to which the authors' (22) K. 8. Cole and R. H. Cole, J . Chem. Phys., 9, 841 (1941). thanks are due. I
I
I
.
..
-
N
(23) S. Glasstone. K. J. Laidler and H. Eyrina. *4Theoryof Rate Processes," MoGraw-Hill Book Co., New York, N.Y.,1941, Chap. IX.
(24) N. Koieumi, to
be published.
i
