Kirkwood treatment

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Figure A2.5.21. The heat eapaeity of an order-disorder alloy like p-brass ealeulated from various analytie treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (seeond approximation) Kirkwood. Eaeh approximation makes the heat eapaeity sharper and higher, but still finite. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 14, figure 13. Copyright (1938) by the Ameriean Physieal Soeiety.

The discussion in the previous section indicated that Nett values would be expected to exceed substantially the actual number of outer shell electrons. Table IV amply confirms this conclusion. Since N appears to the power in the Slater-Kirkwood equation, the deviations are exaggerated. Thus, in making a very similar treatment a few years ago in which slightly different empirical potentials were used, the writer31 found substantially smaller effective N values. For the same reason a relatively crude effective... 

The detailed hydrodynamic treatment of Kirkwood and Riseman yields the following relationship... 

Agarwala SS, Kirkwood JM. Temozolomide, a novel alkylating agent with activity in the central nervous system, may improve the treatment of advanced metastatic melanoma. Oncologist 2000 5 144-151. 

Kirkwood, E., Nalewajko, C., and Fulthorpe, R.R., Physiological characteristics of Cyanobacteria in pulp and paper waste treatment systems, J. Appl. Phycol. 15, 324-335, 2003. 

Fe(5N02phen)3] + aquation in ternary water-Bu OH-polyethyleneglycol (PEG400). " Kinetic patterns for systems of these types have been the subject of theoretical analyses, as in the application of the Savage-Wood Group Additivity principle to [Fe(5N02phen)3f" " aquation in a variety of water-rich binary aqueous mixtures and in aqueous salt solutions " and in the Kirkwood-Buff treatment of preferential solvation of initial and transition states for [Fe(phen)3] + aquation in methanol-water " and for [Fe(gmi)3] " " aquation in Bu OH + water. " ... 

The molecular basis of the Rouse model (and its more sophisticated relatives), however, has remained somewhat obscure. The more complete and realistic treatment of Kirkwood,20 though itself rarely tractable save in a formal sense, could in principle offer a starting point for the derivation... 

Spectral moments can also be computed from classical expressions with Wigner-Kirkwood quantum corrections [177, 189, 317] of the order lV(H2). For the quadrupole-induced 0223 and 2023 components of H2-H2, at the temperature of 40 K, such results differ from the exact zeroth, first and second moments by -10%, -10%, and +30% respectively. For the leading overlap-induced 0221 and 2021 components, we get similarly +14%, +12%, and -56%. These numbers illustrate the significance of a quantum treatment of the hydrogen pair at low temperatures. At room temperature, the semiclassical and quantum moments of low order differ by a few percent at most. Quantum calculations of higher-order moments differ, however, more strongly from their classical counterparts. 

A quantitative theoretical treatment of the dielectric constant, of water and alcohols in terms of hydrogen-bond formation and making use of the gas-molecule values of the electric dipole moment has been published by G. Ostei and J. G. Kirkwood, J. Chem. Phys. 11, 175 (1943). An alternative treatment of water has been made by L. Pauling and P. Pauling (unpublished). 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

Kuzovkov and Kotomin [89-91] (see also [92, 93]) were the first to use the complete Kirkwood superposition approximation (2.3.62) in the kinetic calculations for bimolecular reaction in condensed media. This approximation allows us to cut off the infinite hierarchy of equations for the correlation functions describing spatial distribution of particles of the two kinds and to restrict ourselves to the treatment of minimal set of the kinetic equations which realistically could be handled (Fig. 2.21). In earlier studies [82, 84, 91, 94-97] a shortened superposition approximation was widely used... 

Before discussing mathematical formalism we should stress here that the Kirkwood approximation cannot be used for the modification of the drift terms in the kinetics equations, like it was done in Section 6.3 for elastic interaction of particles, since it is too rough for the Coulomb systems to allow us the correct treatment of the charge screening [75], Therefore, the cut-off of the hierarchy of equations in these terms requires the use of some principally new approach, keeping also in mind that it should be consistent with the level at which the fluctuation spectrum is treated. In the case of joint correlation functions we use here it means that the only acceptable for us is the Debye-Htickel approximation [75], equations (5.1.54), (5.1.55), (5.1.57). 

Therefore, the approximate treatment of the A+B — 0 reaction for charged particles inavoidably requires a combination of several approximations the Kirkwood superposition approximation for the reaction terms and the Debye-Hvickel approximation for modification of the drift terms with self-consistent potentials. Not discussing here the accuracy of the latter approximation, note... 

Maeurer, M. J., Storkus, W. J., Kirkwood, J.M. and Lotze, M.T. (1996) New treatment options for patients with melanoma review of melanoma-derived T-cell epitope-based peptide vaccines. Melanoma Res., 6, 11-24. 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

The starting point of the density-functional treatment is the Kirkwood charging formula. When the solute-solvent interaction potential of interest is v(x), the intermediate states are described as uk(x), where A is the coupling parameter to identify the state. When A = 0, the system is the pure solvent system and o(x) = 0 (no solute-solvent interaction). When A =1, the solute interacts with the solvent at full coupling and u (x) = v(x). The form of M .(x) at 0 < A <1 is arbitrary. The Kirkwood charging formula is an integration over the coupling parameter and is expressed as... 

The idea of correlating momentary multipoles stands behind the customary modeling of dispersion interaction in the form of a multipole expansion, including dipole-dipole (D-D), dipole-quadrupole (D-Q), quadrupole-quadrupole (Q-Q), and so on, terms. We owe the earliest variational treatments of this problem not only to Slater and Kirkwood [34], but also to Pauling and Beach [35], However, when the distance R decreases and reaches the Van der Waals minimum separation, the assumption that electrons of A and B never cross their trajectories in space becomes too crude. The calculation of the intermonomer electron... 

The first theoretical treatment of infrared solvent shifts was given in 1937 by Kirkwood [166] and by Bauer and Magat [167], Eq. (6-8) - known as the Kirkwood-Bauer-Magat (KBM) relationship - has been derived on the basis of Onsager s reaction field theory [80] using the simple model of a diatomic oscillator within a spherical cavity in an isotropic medium of macroscopic relative permittivity r. 

Kirkwood took a more rigorous statistical-mechanical approach in an attempt to incorporate the effect of local ordering. His theory is only valid for rigid dipoles, and it was left to Frohlich to extend the treatment properly to a system of polarisable dipolar molecules. The work is well described in Frohlich s classic text (1949). The final outcome was the following formula,... 

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