Dielectric Behavior of Associated Liquids and Mixtures According to the Kirkwood-Frohlich Statistical Theory H.-G. Li-hmannsroben lnstitut fur Physikalische und Theoretische Chemie, Technische Universitit Brautischweig, 3300 Braunschweig, West Germany The investigation of the dielectric behavior of solutions can give valuable information about molecular properties such as nolarizahilitv and d i ~ o l moment. e In . nhvsical chem" istry laboratory courses the measurement of static dielectric constants. c. is a standard exoeriment for which manv interesting aspeds have repeatediy been reported in this Journal ( I ) . Usually the interpretation of data is based upon the Dehye equation,
where M = molecular weight, d = density, NA = Avogadro's number, eo = permittivity of vacuum, k = Boltzmann constant, T = absolute temperature, and & = molecular electric dipole moment. In eq 1 the contribution of the molecular polarizability, a,can be expressed by the index of refraction, n: 3k-n2)
(e
M - N ~ yl
+ z ) ( n Z+ Z ) d
3 6 3kT
(2)
which often also is referred to as Debye equation. Equations 1 and 2 are essential, as they allow the determination of molecular parameters, a and p, from macroscopic properties, r and n, which are easily measured. Since the Debye theory is applicable only when the concentration of polar molecules is low, i t excludes a large variety of interesting phenomena occurring in pure polar liquids and in mixtures with high concentrations of polar molecules. The dielectric theoryof such systems has been developed by Onsager. But experiments for the investigation of the dielectric behavior. that are suitable for nhvsical . " chemistrv laboratory courses, have rarely bern reported ( 2 ) .hforeo\,er, these ex~erimrntsseem to be limited to liauids that do nor show specific interactions such as hydrogen bonding, association, complex formation, etc. Evidently, these phenomena have not been included because the appropriate dielectric theories are rather complex and are not easily cast into lucid formulas suitable for teaching purposes. In this work statistical approaches to dielectric theories, to which the names of Kirkwood and Frohlich are attached, are studied by introducing suitable simplifications. The aim is to provide for physical chemistry students a t the graduate level an access to involved dielectric phenomena, which are areas of active research today (3). As prototypical process, association is investigated with pure water and cyclohexanel-propanol mixtures as model systems. The experiments reported here require only a standard apparatus available in most laboratory courses. Theoretical Background A significant advance in dielectric theory was made by Onsaaer's evaluation of the local field actina on a molecule A with; permanent dipole moment. ~ictorialiy,the field of the oolarizes the surrounding molecules, even if they posdipole . sess no permanent dipole moment. The resulting pola;ization of the environment produces a field a t the location of A, which superimposes with the external field. The Onsager equation (4-6),
is derived under the assumption of a homogeneous environment and thus does not allow for interactions that induce some kind of order in the vicinity of A. For very small values of r the left-hand sides of eqs 2 and 3 are almost equal, and dipole moments may be obtained from the Dehye equation. But for r = 4 the deviation of p values is approximately 25% and only the Onsager equation gives correct dipole moments. I t is interesting to note that, although eqs 2 and 3 require n of the infrared region, .. . which often is difficult to measure, wme approximations made in thp derivation oieq 3 cancel out i f n from the visible part of the spectrum is used (6). The next step in the improvement of dielectric theories is the inclusion of short-range intermolecular forces by the statistical approach of Kirkwood and Frohlich (KF). These forces induce a hindrance of molecular rotation that can be accounted for by the introduction of the K F correlation factor g (4-6), where the notation p ~ is p used to distinguish dipole moments of the K F theory from those of the Debye and Onsager treatment (p). The factor g is defined where Z is the number of nearest neighbors and (cos 7 )is the mean cosine of the angles between the dipole moments of neighboring molecules. The KF equation of dielectric theory,
is a generalization of Onsager's equation (eq 3). In the ahsence of short-ranae interactions. molecular rotation is unhindered and, with (cos y ) = O,eq 3 is recovered. With p determined a t low concentrations (ideally a t low pressures in the gas phase), in liquids g can be obtained from measurements of r and n. Evidently g is an important key to the understanding of correlated mutual orientation in liquids. Results and Discussion
Temperature Dependence of the Dielectric Constant of Water A major achievement of the KF theory was the explanation of the high dielectric constant of water: c(HzO) = 78 at 25 'C.' For the interpretation of the temperature dependence of c(H20) i t is useful to introduce some simplifications. Firstly, r(Hz0) >> n2. Therefore, n2 has been neglected
'
If not stated otherwise. literature data of f .. n.. d. . ,u.. etc... are from ref 7. For water, temperaturedependent values of dand n are also obtained from ref 7.
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Figure 1. Temperature dependence of dielectric constant ol water ( 0 ,student measurements:-, data from ref 8). in the nominator of the left-hand side of eq 6. Secondly, since n2 = 2 is a good approximation for many liquids, this has also been assumed for water.2 With these assumptions, eq 6 reduces to the simple relation (6),
Typical experimental results obtained by a student group together with data from ref 8 are shown in Figure 1. The evaluation of the experimental data according to eq 7 C2 m2. Here the error eives eu2 = (1.10 f 0.15) X yndicates the deviation of the results of about 20 groups over a one-vear oeriod. If the erouos were orovided witha calibration cirve the error reduces fo 5%.w i t h the gas phase value C m, the of the dinole moment of water. rr = 6.1 X correlatidn factor was obtained ti be: g = (2.95 f 0.40): For camoarison. the students often also used ea 3 to determine yu-, bur nosignificnnt improvemenr oithe results was found. 'l'hi.; \.nlidates the a s s u m ~ t i m smade in the deri\.ation of eq 7. In their original work Oster and Kirkwood (9) proposed (cos r ) = cos2
(g)
= 0.371
(8)
where 11. = 105' is the H-0-H bond anele in the water molrcuie determined spectrosropically. [A &htly different mcrdel is introduced in ref 5 ) . Frum this the itudenth cnlculated the number of n e a r e ~ t n e i ~ h b oto r s be Z = 5.3 f 1.1. The results were compared with data from the literature. For example, in ref 9 Z = 4.4-4.9 and g(observed) = 2.752.49, and g(calculated) = 2.63-2.82 has been reported. Frequently the students also discussed their results in relation to tetrahedral bond symmetry and hydrogen honding; temperature dependence of Z and g; and the relation between radial distribution functions, for example, those obtained from X-ray diffraction, and the simple concept of nearest neighbors (10). Dielectric Behavior of Cyclohexane- 1-Propanol Mixtures For the investigation of binary mixtures, the KF equation (eq 6) has to he extended. However, the general treatment leads to complicated expressions that are not easy to use in a laboratory course (4). In the following, a mixture of a nonpolar (with xo, Mo, do, and no as mole fraction, molecular weight, density, and refractive index, respectively) and a polar component (x1,M1,dl, nl) is considered. As before n i = n12 = 2 is assumed. Moreover, the density of the mixture, d, 680
Journal of Chemical Education
Figure 2. Dielectric behavior of cyclohaxane-I-propanol mixtures at 20 ' C : g as function of the I-propanol niole fraction(e. X, student measurements;-, average of about 20 measurements; 0 ,data from the table).The inset shows the behavior at cyclohexane-I-butan01mixtures at 40 ' C (13. Dielectric Constants ot Cyclohexane-1-Propanol Mixtures, 6,. for Various Mole Fractions of 1-Propanol ( x , ) at 20 "C
0 0.014
0.028 0.055 0.078 0101 0183
0.375 0.483 1.0
2.01 2.03 2.04
2.09 2.15 2.22 2.66 5.20 7.52 19.3
-
0.0448 0.0596 0.133 0.219
0.319 0.908 3.815 6.25
18.2
5990 2980 1510 1050 809
436 200 150 60
I
0.73 0.48 0.54 0.63 0.70 1.09 2.10 2.56 2.99
m e correlation factor g has been evaluated with the 1-propanol dipole momem p, = 5.57 X 10-lOcm ( I ) .
is taken to be constant. It has to be emphasized that the first assumption is valid for many liquids, whereas the second condition can onlv be fulfilled if d = d l = d7 and if the cumponrnts iorm ~ p p r o x i m a t eideal ~ ~ mi&ures. With thew simr~lificationsthe KF equation for hinars mixtures is found to be"
where E , is the dielectric constant of the mixture. Since eq 9 was to be used in the whole range of 0 < xl 5 1, r , cannot assume to be larger than n2. The conditions introducdd for the derivation of eq 9 are fulfilled by cyclohexane (at 20 "C:do = 0.779 g/mL and no2= 2.04 a t N a D line) and 1-propanol (dl = 0.804gImL and n12= 1.92). The densities of the binary mixtures do not significantly deviate from ideality ( l l ) , and for the evaluation of the data an average density of d = 0.791 was assumed. The error induced by this procedure was less than 2%. A typical set of experimental data is presented in the table. For the evaluation of g, the students determined the dipole moment of 1-propanol, fil, in cyclohexane with the Debye equation for r , extrapolated to infinite dilution. This 2For water rP = 1.779-1.762 (at Na D line) in the temperature range of 15-60 O C . With the assumptions made, eq 9 is derived from ref 4, p 261, eq 6.208. which is the general KF equation in binary mixtures.
nomena. Due to the complexity, however, graduate-level experiments elucidating the K F dielectric theory seemingly have not been reported. In this work simplifications are introduced that allow the application of the theory to the dielectric behavior of associated systems without reducing the validity of the conclusions. Results obtained from elementary experiments, performed in our physical chemistry laboratory course, are presented.
antiparallel
parallel association
Figure 3. Possibleorientationofalcohol dimersandthe resulting gvalues(i3).
procedure has been described in detail ( l e ) and does not need to he repeated here. The measured value of pl was (4.7 f 0.5) X 10POCm, which is slightly below the gas phase value from the literature: u~ = 5.6 X Cm. The dependence of g on the mole fr&on of 1-propanol measureb by two student ZrOUDS is shown in Figure 2. The line depicts an average of ali student measureme&. While there issignificant scatter of data close to the minimum, the qualitative behavior of g is correctly determined. The agreement with results in the literature, obtained for similar systems from verv soohisticated data evaluation. is excellent (cf. Fie. 2) (4. 12): w i t h these results the students discussed the association of alcohols within simole conceots of multimer formae In dilute tion, as exemplified for dfmers in ~ i ~ u 3r (13). solution g is less than 1 and antinarallel association of the alcohol n~oleculesis predominant. .At higher concentratims of 1 -.u r o.~ a n othe l number of cumolexes with rmrallel diunle moments increases (1 5 g 5 2). In pure l-propanol, with g = 3. comolex formation includes hizh-order multimers with large dipole moments. The students often compared this behavior with other interesting cases, occurring for example in carhoxylic acids and polymers (4). Concluslons In teaching physical chemistry, dielectric theories are particularlv well suited for familiarizine students with the development of theoretical concepts. 'fhe approaches of Clausius-Mosotti. Dehve. . . Onsaeer. - . and Kirkwood-Frohlich (KF) allow the interpretation of increasingly intricate phe-
Experlrnental Condltlons Dielectric constants of the solutions were measured with a WTW 03 "Dipole Meter"'. Three condenser cells for various ranges of r were available: MFL-1 (e 5 7), MFL-2 (c = 7-21) and MFL-3 ( r = 21-88). Calibration of the dipole meter scale was performed by the students with suitable neat reference liquids and values of e from the literature. All measurements were performed at 1.8MHz. According to the manufacturer,the readings of the dipole meter had an accuracy of 0.1%;absolute values of < were subject to the precision of the calibration curve. Temperature control of the cells was achieved by circulating water from a thermostat. From 10 "C to 70 'C a stability of 1.0 OC was achieved. Measurements of the cyclohexane-1-propanol mixtures were performed at 20 OC, while water was investigated in the range
-.
is-fin "" oc A"
Water was distilled three times before use; cyelohexane and 1propanol were from Merck (pa.) and were used as received. Acknowledgment Thanks are due to the students of the Advanced Physical Chemistry Laboratory Course, winter semester 1985186 and summer 'emester 1986, a t our institute for their sometimes enthusiastic willingness to test various experimental approaches, and to J. Mundt and S. Schwarzer for valuable technical assistance. I am grateful to R. Lacmann for stimulating my interest in this work. 1, See, for example, (a1 Janini, G. M.; K8trib.A. H. J . Chem. Edue. 1983,60, 1087: (h) Bonills,A.:Vasroa,B. J,Chsm.Educ. 1977,54,130:(e)Kurh,S.R.;Anderson,O.T.: Willeford, B. R., Jr. J . Chsm. Edue. 1971.54, 181; id) Coe, D.A,; Nibler, J. W. J . Chem. Educ. 1913,50,8Z: ie1 Thompson, H. B. J. Chem. Educ 1966,43, 66: IO Moffat. J. B.; J Chem. Edue. 1966.13. 74. (81 Conde, J. P.; Moura-Ram-, J. J. J . Chem. Educ. 1966,63,823. 2. Daniels, F. et a]. Ezp~rimentoiPhysical Chemistry, Mecraw-Hill: N e w vork, 1970. 3. Mandrl,M.:Odijk.T.Ann.Rau.Phyr. Cham. 1985,35,75. 4. Bbttcher. C. J. F. Theory "/Electric Poiorilotion: Elsevier: Amsterdam, 1973. S. Chehwaki, A.Diei~ctricPhyairr: Elrevier: Amsterdam, 1980. 6. Korturn. G.Lehrbvrh iOrEBktrochemir; Verlsg Chernie: Weinheim, 1966. 7. Handbook of Chemistry and Physics. 66th ed.: CRC: Cleveland, OH. 1985. 8. Eirenberg, D.: Kauzmann. W. The Sfrurfure and Proparties o/ Water; Clarendon: oxford, 1969. 9. 0ster.G.; Kirkwood, J.G. J . Chem. Phys. 1943,11,175. 10. Moore. W. J. Physical Chemistry; Longmsn: London. 1972. 11. Lacrnann. R. In Numwirol Daro andFunctionolReiolionrhipa in Science ond TechnoloEy,N~mSaries;Hellwege,K.H.. Ed.:Springer:Bedin, 1977:Vol. 1,Parts. p220. 12. Mecke.R.: Reuter,A.Z.Naturlorsch. 1949.4a.368. 13. Go1d.P. 1;Perrine. R. L. J . Phys. Chrm. 1967,71,4218.
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