Kirkwood correlation parameter

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... 

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 ... 

Table 16-1 shows results for the dielectric constant e(0), Kirkwood -factor gK, and the static dipole cross-correlation parameter g° = ( M(0) 2) /(Np ) — 1 where M(f) = IFit) is the system s collective dipole at time t, for a selected set of thermodynamic states. The experimental values for e(0) are shown within parentheses. The overall trend of these quantities with density and temperature is consistent with the expectation of a higher degree of dipolar correlation at higher densities and/or lower temperatures. At liquid-like densities (states 10-12), where polarizability effects are known to be important, the simulated model underestimates e(0), a feature common to most non-polarizable water models. Given the error bars and differences in thermodynamic states, our estimates for e(0) for states 10-12 are... 

If one compares the rate constants for the same Menschutkin reaction with Kirkwood s parameter in thirty-two pure aprotic and dipolar non-HBD solvents [59, 64], one still finds a rough correlation, but the points are widely scattered as shown in Fig. 5-11. 

Kirkwood, in his statistical theory of dipolar dielectrics, derived the following angular correlation parameter ... 

Simple Liquids. The correlation parameters (151) and (152) are accessible to numerical calculation for simple models of radial interaction between the molecules. In a number of cases, evaluations for simple liquids can be usefully made using the Kirkwood model of hard spheres of diameter d and volume v = Trd lS, when... 

Nishikawa, K., Hayashi, H., lijima, T. (1989). Temperature dependence of the concentration fluctuation, the Kirkwood-Buff parameters, and the correlation length of tert-butyl alcohol and water mixtures studied by small-angle X-ray scattering. Journal of Physical Chemistry, 93, 6559-6565. 

The above examples illustrate that continuum models such as the Kirkwood model are reasonably successful in describing the static permittivity, provided one has an independent means of estimating the correlation parameter Unfortunately, these estimates are available for only a few polar solvents, so that gK must be considered an independent parameter. The version of Kirkwood s theory presented here only considers orientational polarization. When distortional polarization, that is, the effect of molecular polarizability, is included, interpretation of experimental results is less clear. Since the approach taken here involves continuum concepts, it is necessarily limited. In the following section, a simple model based on a molecular description of a polar liquid is presented. 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

There are other striking results that emerge from the use of the SSOZ equation with any of the closure equations given above to calculate the dielectric constant and the Kirkwood angular correlation parameters. We consider some of these in the next section. 

Another measure for order in a liquid is the Kirkwood dipole orientation correlation parameter (KDOCP) g, but this pertains only to dipolar liquids ... 

A further quantity that describes the structuredness and self-association of solvents is the Kirkwood dipole angular correlation parameter. This parameter is g = l + Z cos0, where Z is the number of nearest neighbors a solvent molecule has... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

Ab initio SCRF/MO methods have been applied to the hydrolysis and methanol-ysis of methanesulfonyl chloride (334). ° The aminolysis by aromatic amines of sulfonyl and acyl chlorides has been examined in terms of solvent parameters, the former being the more solvent-dependent process.Solvent effects on the reactions of dansyl chloride (335) with substituted pyridines in MeOH-MeCN were studied using two parameters of Taft s solvatochromatic correlation and four parameters of the Kirkwood-Onsager, Parker, Marcus and Hildebrand equations. MeCN solvent molecules accelerate charge separation of the reactants and stabilize the transition... 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

As in previous Chapters, for practical use this infinite set (7.1.1) has to be decoupled by the Kirkwood - or any other superposition approximation, which permits to reduce a problem to the study of closed set of densities pm,m with indices (m + mr) 2. As earlier, this results in several equations for macroscopic concentrations and three joint correlation functions, for similar, X (r,t),X-s r,t), and dissimilar defects Y(r,t). However, unlike the kinetics of the concentration decay discussed in previous Chapters, for processes with particle sources direct use of Kirkwood s superposition approximation gives good results for small dimesionless concentration parameters Uy t) = nu(t)vo < 1 only (vq is d-dimensional sphere s volume, r0 is its radius). The accumulation kinetics predicted has a very simple form [30, 31]... 

FA of data matrices containing 35 physicochemical constants and empirical parameters of solvent polarity (c/ Chapter 7) for 85 solvents has been carried out by Svoboda et al. [140]. An orthogonal set of four parameters was extracted from these data, which could be correlated to solvent polarity as expressed by the Kirkwood function (fir — l)/(2fir + 1), to solvent polarizability as expressed by the refractive index function (rfi — + ), as well as to the solvent Lewis acidity and basicity. Thus,... 

The parameter g is the correlation factor first introduced by Kirkwood (1939) to take account of local order ... 

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