Kirkwood definition

There is no limitation for the use of Kirkwood s superposition approximation (2.3.62) in the integral terms of equation (7.1.27) since there is a single defect in the recombination sphere of dissimilar defect. For example, a product a( r2 — f[ )piti(r2 f( t) in a line with definition (7.1.3) could be replaced by <7i,i(r2 rj <). Therefore, we have... 

In deriving (8.3.11) we took into account the fact that despite the reaction rate is still defined through the integral (4.1.19), its magnitude depends greatly on the equation for the correlation function for dissimilar particles. Making use of the definition of the functional (5.1.5), let us write down equation (8.3.8) (in usual dimensional units) modified by the Kirkwood... 

These equations can be expressed in terms of the chemical potentials of the salts when the usual definition of the chemical potentials of strong electrolytes is used. The transference numbers may be a function of x as well as the molality. Arguments which are not thermodynamic must be used to evaluate the integrals in such cases (see Kirkwood and Oppenheim [33]). One special type of cell to which either Equation (12.112) or Equation (12.113) applies is one in which a strong electrolyte is present in both solutions at concentrations that are large with respect to the concentrations of the other solutes. Such a cell, based on that represented in Equation (12.97), is... 

However, Eqs. 3 and 5 are different equations even though they are based on the same definition of the preferential binding parameter and have the same theoretical basis the Kirkwood-Buff theory of solutions. To make a selection between Eqs. 3 and 5 a simple limiting case, the ideal ternary mixture, will be examined using the traditional thermodynamics, and the results will be compared to those provided by Eqs. 3 and 5. 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

One of the important methods of developing a model for the potential energy of the liquid system is that based on integral equations. These include the Kirkwood integral equation, the Born-Green-Yvon equation, and the Omstein-Zernike equation. The last approach leads to the definition of the direct correlation function c(fi2). It is the approach which is most frequently used and is the one which is considered here. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

As was shown by Kirkwood,13 if dynamical measurements are carried out on polymers in dilute solutions, a new characteristic length appears, the hydro-dynamic radius ftH. By definition... 

An approximate expression for the Kirkwood cor relation factor can be derived by taking into account only the near-est-neighbors interactions. In this case the sphere is reduced to eontain only the Jth molecule and its z nearest neighbors. For this definition it is possible to derive the following relationship for the parameter g ... 

Equation (3.4.1) follows directly from the definition of the process of solvation (see also Appendix G). The expression for the solvation, Gibbs or Helmholtz energy in terms of the process of inserting a particle at a fixed position is quite old, probably due to Kirkwood (1935) and later used in the scaled particle theory [see Sec. 3.8 and also Hill (1960) and Widom (1963,1982)]. 

FIGURE 1.2 The Kirkwood—Buff inversion approach for obtaining KBIs from the available experimental data. See Prolegomenon for symbol definitions. (Ploetz, E. A., and Smith, P. E., 2011, Local Fluctuations in Solution Mixtures, Journal of Chemical Physics, 4, 135. With permission of the American Institute of Physics.)... 

The paper by Kirkwood and Buff is quite remarkable. It is only four pages long and only two of those four contain the important parts of the theory. It does this by giving only definitions and results and leaving the reader to fill in all of the intermediate steps. Most of these are straightforward, but there are several tricky points, which we shall discuss below. 

Equation (IIIB-83) is identical to (MFK-Equations 20, 21). It is also identical to Kirkwood s (1937) equation. Using the dipole approximation to ViQaiiot nd the definition of a polarizability tensor (Equation IIIB-68), we see that (IIIB-83c) can be written formally as a function of at (v) and a/ (v). These are tensors for groups t and j whose principal values and principal directions are frequency dependent. If we know the a s at each frequency we know the ji s and we can use (III B-83c) as it stands. 

In this section we first define the PS in terms of the Kirkwood-Buff integrals for a three-component system a solute S and a two-component solvent, say of A and B. Then we generalize the definition of PS for a two-component system, of A and B only. 

Computer simulation has been used to obtain sudi local thermodynamic functions as the energy density, < (z), the transverse component of Irving and Kirkwood s definition of the pressure tensor.s" Pr(z), and hence the height of the related surface of tension in a planar interface.s° The essential arbitrariness of these calculations has been discussed in 4.3 and 4.10. 

Evaluating the mean square moment of a sphere (Eq. (31)), and using the definition of the correlation factor above gives the anisotropic version of the Kirkwood-Froh-lich equation ... 

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