Kirkwood correlation

Initiator decomposition studies of AIBN in supercritical C02 carried out by DeSimone et al. showed that there is kinetic deviation from the traditionally studied solvent systems.16 These studies indicated a measurable decrease in the thermal decomposition of AIBN in supercritical C02 over decomposition rates measured in benzene. Kirkwood correlation plots indicate that the slower rates in supercritical C02 emanate from the overall lower dielectric constant (e) of C02 relative to that ofbenzene. Similar studies have shown an analogous trend in the decomposition kinetics ofperfluoroalkyl acyl peroxides in liquid and supercritical C02.17 Rate decreases of as much as 30% have been seen compared to decomposition measured in 1,1,2-trichlorotrifluoroethane. These studies also served to show that while initiator decomposition is in general slower in supercritical C02, overall initiation is more efficient. Uv-visual studies incorporating radical scavengers concluded that primary geminate radicals formed during thermal decomposition in supercritical C02 are not hindered to the same extent by cage effects as are those in traditional solvents such as benzene. This effect noted in AIBN decomposition in C02 is ascribed to the substantially lower viscosity of supercritical C02 compared to that ofbenzene.18... 

Form factor of the hat-curved model Normalized concentration of molecules Kirkwood correlation factor Steady-state energy (Hamiltonian) of a dipole Dimensionless energy of a dipole Moment of inertia of a molecule Longitudinal and transverse components of the spectral function Complex propagation constant Elasticity constant (in Section IX)... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

Up to now we neglected interaction between the particles of a medium. To take it roughly into account, we have introduced in Eq. (30) the Kirkwood correlation factor g. The latter, in general, differs from 1, since the total dipole... 

The results of the calculations of the spectra are illustrated by Figs. 18 and 19. The first figure refers to the temperature T = 133 K, which is near the triple point (131 K for CH3F). In this case the density p of a liquid, the maximum dielectric loss t j( in the Debye region, and the Debye relaxation time xD are substantially larger than those for T = 293 K (the latter is rather close to the critical temperature 318 K) to which Fig. 19 refers. The fitted parameters are such that the Kirkwood correlation factor is about 1 at T = 293 K. 

Here g is the Kirkwood correlation factor (146), which is determined by the static permittivity ss, permittivity irx at the HF edge of the librational band and by the reduced concentration of the dipoles G. The total number of free parameters of our model is now six ... 

In our case of 1-1 electrolytes we have the following relations for the concentrations N 1 = N = Nmn. The first term in Eq. (413a) is calculated by using the formulas presented in Section IV. However, generally speaking, the Kirkwood correlation factor now should be determined with account of ionic static permittivity as... 

In view of the studies [75] of aqueous NaCl solutions by using molecular-dynamics simulation, one may suggest that our models are applicable61 only at low salt concentrations Cm, for which the Kirkwood correlation factor g could be calculated from Eq. (414d), where the ionic contribution Ae on(0) is not involved. 

However, the latter formula is not more applicable, if xlaa is rather long and/ or Cm is rather high, so that the zero-frequency ionic contribution Aefon(0) to permittivity is noticeable in comparison with the static permittivity es of the solution. We note that the Kirkwood correlation factor g is used for calculation of the component p in Eq. (387). Thus, even in our additivity approximation the solvent permittivity )jip is determined in this case by concentrations of both solution components. This complication leads to a new self-consistent calculation scheme. 

In order to describe the correlations between the orientations (and also between the positions) of the z th molecule and its neighbors, Kirkwood introduced a correlation factor g, which may be written g = Sjlj (cos 0,y), where 0,y denotes the angle between the orientation of the zth and the /th dipole [7]. An approximate expression for the Kirkwood correlation factor can be derived by taking only nearest-neighbor interactions into account. It reads as follows ... 

We now turn to discussion of the computer dynamical procedure, which shows great promise for calculations of static permittivity. The model of Rahman and Stillinger contains sufficient information in the computer program to enable the mean dipole moment A of a sphere of known radius to be calculated at a known temperature. This moment is related to the Kirkwood correlation parameter g by equations (9) and (10), with g defined as in equation (10). 

Values of Xs and have been estimated for the polar solvents considered in this chapter and are summarized in table 4.4. Also included in this table are values of the Kirkwood correlation parameter g. It is defined in the MSA model as... 

Solvent Polarization Parameter ks Stickiness Parameter h Molecular Polarizability lO otp/nm Kirkwood Correlation Parameter... 

Note, finally, that setting frequency x to zero in Eq. (5), we may relate the static librational susceptibility xor> 5 to the Kirkwood correlation factor gor as... 

Guggenheim s characteristic free energy in electric fields, transformed Gibbs free energy Conversion factors Kirkwood correlation factor Ionic strength (mol dm )... 

In a medium of polar molecules specific intermolecular interactions such as, for instance, H-bridges in water may occur. The effect of this property is accounted for by the Kirkwood correlation factor... 

A topic of abiding interest is the issue of characterizing the order in liquids which may be defined as the entropy deficit due to preferential orientations of molecular multipoles relative to random orientations (orientational order) and nonuniformly directed intermolecular forces (positional order). Phenomenologically, two criteria are often claimed to be relevant for deciding whether or not a liquid is to be viewed as ordered the Trouton entropy of vaporization or Trouton quotient and the Kirkwood correlation factor gK. Strictly speaking, however, both are of limited relevance to the issue. 

Any molecular association, especially the dipole-dipole one, strongly in-fiuences the thermodynamic properties of mesophases. Local dipole correlation can be quantified by measurements of the Kirkwood correlation factor defined as [65]... 

Specific pairwise dipole-dipole interactions can be accounted for by introducing the Kirkwood correlation factor gi, such that the mean square dipole moment is replaced by an effective mean square moment defined by ... 

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