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PHILLIP I. GOLDAND RICHARD L. PERRINE
The Kirkwood-Frohlich Correlation Parameter and the Structure of Alcohol and Carbon Tetrachloride Mixtures1
by Philip 1. Gold2 and Richard L. Perrine Department of Engineering, University of California, Los Angeles, California 90004 (Received March 8, 1967)
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The Kirkwood-Frohlich equation provides a method for evaluating the so-called correlation parameter, of which the deviation from unity is a measure of the degree of hindered molecular rotation of the alcohol molecules arising from short-range intermolecular forces. The correlation parameter is found to be greater than 1 for the pure alcohols and is found to decrease to less than 1 in mixtures dilute in alcohol. This indicates that while the pure alcohol is highly coordinated into chainlike structures, a t least an appreciable number of dimer and/or trimer complexes must form closed or cyclic chains with reduced dipole moments. The correlation parameter is found to be a sensitive indicator of the steric effects due to size-shape changes found in the homologous alcohol group.
Introduction A number of progressively more detailed theories have been developed to describe the dielectric behavior of fluids the molecules of which possess a permanent dipole moment. The simplest theories, due to ClausiusMosotti and D e b ~ e ,neglect ~ intermolecular forces. These have yielded satisfactory results only for gases at low pressures. The Onsager theory4recognizes that, when a molecule having a permanent moment is surrounded by other molecules, the field of this dipole polarizes the surrounding molecules, since they have polarizability even if no permanent moment. This polarization of the environment of a molecule will give rise to a field at the location of the molecule. The theory is limited by the assumption that the molecule occupies a spherical cavity in the dielectric. .A more important limiting factor is the treatment of the entire surroundings of each molecule as a homogeneous continuum, making no distinctions between interactions of closely neighboring molecules and those relatively far apart. Just this distinction is needed for associated liquid^.^ The treatment can be extended to consider binary mixtures. Kirkwoods and later Frohlich7.8 showed that, in order to explain the obvious discrepancies between Onsager’s theory and experimental results, some way would have to be devised to separate the immediate neighbors of a molecule from the continuum of Onsager’s model. I n The J o u r a l of Physical Chemistry
this way the effects of hindered molecular movement resulting from short-range intermolecular forces could be determined. Frohlich’s Statistical Approach The Frohlich method, which uses classical statistical mechanics, is given in detail in the references cited. I n this paper it will suffice to present the essential character of the theoretical description. Frohlich7s8has set up a model in which the liquid is considered as a continuous medium of dielectric constant m2,where n* is the refractive index of the liquid considered. I n this medium are imbedded dipoles containing moment ;‘. It is convenient to consider a spherical specimen of volume V divided into n unit cells, so that N = n/V is the number of unit cells per unit vol(1) Taken in part from a Ph.D. dissertation submitted by P.I. Gold to the Graduate Division, University of California, Los Angeles, Calif.. Jan 1965. (2) American Chemical Society Petroleum Research Fund Fellow, 1961-1964. (3) P. Debye, “Polar Molecules,” Dover Publications, New Pork. N. Y., 1945. (4) 1,. Onsager, J . A m . Chem. Soc., 58, 1486 (1936). (5) C. Bottcher, Phyaica, 6 , 59 (1939). (6) J. G. Kirkwood, J. Chem. Phys., 7 , 911 (1939). (7) H.Frohlich, “Theory of Dielectrics,” Clarendon Press, Oxford, 1950,pp 36-53. (8) H. Frohlich, Trans. Faraday Soc., 44,238 (1948).
THEKIRKROOD-FROHLICH CORRELATION PARAMETER
ume. If each unit cell contains just one dipole, ;‘, then p* can represent the moment of the entire spherical region due to polarization by one of the unit cells. For this medium the observed dielectric constant, e, should be given by the relation
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(t
- 1)(2€ 3E
+ 1) M_ d
--
(n*’
-
It is important to note that the term ’; ;*‘ refers to the situation encountered in the absence of an external field. If there were no tendency for a molecule to hinder the movement of its neighbors, one would find that the dipole ‘;would have no-.+ net orienting effect on its neighbors, making *; = p. Therefore, one would obtain: - * p * p* 7p2. Thus one concludes that the difference between p*‘;* and p2 represents the effect of hindering torques which exist between molecules. It is further demonstrated by Frohlich that the same moment *; is containing in any sphere surrounding the dipole ;’, regardless of the size of the sphere. It follows, then, that any interactions leading to deviations between and ;*‘ must occur between closely neighboring dipoles. The problem now remains to calculate ‘;*. Since all contributions to ;*‘ are due to orientation, ;*‘ is the average moment due to the permanent dipoles of a spherical region if one of its dipoles, p , is kept in a fixed direction. From classical statistical mechanics, this definition yields
+ 2)(2t + 1)
1
for a pure dipolar compound. M is molecular weight, d is density, N Ais Avogadro’s number, and k is the Boltamann constant. n* refers to the refractive index measured at frequencies high enough so that orientational polarization effectively has ceased. For the case of no hindered molecular rotation (g = l ) ,the result is identical with that of Onsager. Finally, Frohlich’s result can be extended to the case of binary mixtures in much the same way as the Onsager treatment (E
- 1)(2t + 1) _ M -3e
(n*%
d
+ 2 dl
;
z.s* = /2[1
f
I= +
n’J (cos y)e-w’kT dwdv e-w/dT dwdv p2[1
n’ Cosy] (2)
where n’ is the number of nearest neighbors to the given dipole, y is the angle between the dipole moments of an arbitrary pair of molecules, and W is the potential of the average forces acting on them. The integrations are carried out over all relative orientations of the dipoles. The term [l n’ cos y ] represents the effect of hindered rotation and is called the correlation parameter, g, so that, ‘;.’;* = gp2. It can be shown that for the Frohlich model the moment is related to the moment of a free molecule, Go, by
+
3
9kT
where the subscripts 1 and 2 refer to the polar and nonpolar components, respectively. This equation takes explicit consideration of the effects of hindered molecular rotation due to hydrogen bonding on the dielectric behavior. It is obvious from this equation that the correlation parameter, which is a direct measure of these effects, is a function of both temperature and concentration. To show how the correlation parameter may vary, consider the changes, according to eq 2, which follow different modes of association. If only dimers are formed, g near zero indicates strong antiparallel association, while a value of g near 2 indicates strong parallel association R
R
R antiparallel association, 0 = 0 parallel association. 1
(3) Inserting eq 2 and 3 into 1 and rearranging, there obtains
