Pair correlation functions pressure

The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

LDA and HDA were interpreted to be similar to two limiting structural states of supercooled liquid water up to pressures of 0.6 GPa and down to 208 K. In this interpretation, the liquid structure at high pressure is nearly independent of temperature, and it is remarkably similar to the known structure of HDA. At a low pressure, the liquid structure approaches the structure of LDA as temperature decreases [180-182]. The hydrogen bond network in HDA is deformed strongly in a manner analogous to that found in water at high temperatures, whereas the pair correlation function of LDA is closer to that of supercooled water [183], At ambient conditions, water was suggested to be a mixture of HDA-like and LDA-like states in an approximate proportion 2 3 [184-186],... 

In the liquid phase, calculations of the pair correlation functions, dielectric constant, and diffusion constant have generated the most attention. There exist nonpolarizable and polarizable models that can reproduce each quantity individually it is considerably more difficult to reproduce all quantities (together with the pressure and energy) simultaneously. In general, polarizable models have no distinct advantage in reproducing the structural and energetic properties of liquid water, but they allow for better treatment of dynamic properties. 

However, there is no doubt that from the 1980s on, a very hopeful type of development has been taking place in ionic solution theories. It is the correlation function approach, not a theory or a model, but an open-ended way to obtain a realistic idea of how an ionic solution works (Fig. 3.58). In this approach, pair correlation functions that are experimentally determined from neutron diffraction measurements represent the truth, without the obstructions sometimes introduced by a model. From a knowledge of the pair correlation function, it is possible to calculate properties (osmotic pressure, activities). The pair correlation function acts as an ever-ready test for new models, for the models no longer have to be asked to re-replicate specific properties of solutions, but can be asked to what degree they can replicate the known pair correlation functions. 

Estimate pressure—volume—temperature (PVT) relations, cohesive energy densities, pair correlation functions, X-ray scattering curves, elastic constants, and other properties. 

The simulation of the liquid has given a vaporization energy underestimated by 7%, while pressure appears to be overcorrected with respect to MCY at 305 K (P =-1180 + 470atm). Experimental specific heat is well reproduced, as pair correlation functions, apart from the usual shift to larger distance of the first peak of the 0-0 g(r). The effective dipole moment is 2.8 D and diffusion coefficient (r> = 2.5 0.110" cmVs) compares well with the experimental data at 25 C, = 2.4-10" cm /s). Experimental is reproduced... 

Y.E. Gorbaty and Y. Demianets, The pair correlation functions of water at pressures of 1000 bar in the temperature range 25-500 C, Chem. Phys. Lett., 100 (1983) 450-454. 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

The compressibility equation involves the radial distribution function even when the system consists of nonspherical particles. We recall that previously obtained relations between, say, the energy or the pressure, and the pair correlation function were dependent on the type of particle under consideration. The compressibility depends only on the spatial pair correlation function. If nonspherical particles are considered, it is understood that g(R) in (3.109) is the average over all orientations (3.105). In the following, we shall remove the bar over g(R). We shall assume that the angle average has been taken before using the compressibility equation. 

We choose a two-component system for which the pressure (or the total density) is sufficiently low such that the pair correlation function for each pair of species has the form (see section 2.5)... 

The pair correlation function is important for various reasons. For one thing, a knowledge of the pair correlation function is sufficient to calculate various thermodynamic quantities (particularly the potential energy and the pressure), assuming that the total interaction potential is a sum of pairwise interactions [1, 189, 190]. For another, the structure factor, which is the Fourier transform of the pair correlation function. 

In the site-site representation, the virial pressure involves higher spherical harmonic coefficients of the site-site pair correlation functions. The difficulties involved in the standard virial route to the pressure in polymer systems have been well illustrated in recent work. ... 

Here u. j(r,T,P) is the short-range potential for ions, and Eq is the dielectric constant of the solvent. The solvent averaged potentials are thus actually free energies that are functions of temperature and pressure. The thermodynamic properties calculated from the pair correlation functions are summarized below. 

The evaluation of the pair correlation functions of both the solvent molecules and the solutes is feasible but troublesome. For electrolyte solutions, however, averaging over the solvent effects yields reliable approximations, as shown by McMillan and Mayer, who considered the solution in osmotic equilibrium with the pure solvent (Fig. 2). They stressed that solutes can be treated as an imperfect gas, provided that one uses the potential of mean force at infinite dilution. The calculation yields the osmotic pressure Tl = P — Pq (see Fig. 2) from the virial equation [Eq. (69)] in terms of the forces among the particles for the solution state (p, T). The independent variables of... 

Gorbaty, Y. E. Demyanets, Y. N. (1983) X-ray Diffraction Studies of the Structure of Liquid and Supercritical Water at High Temperatures and Pressures. II. The Molecular Density Radial Distribution Functions and the Paired Correlation Functions, Journal of Structural Chemistry (Engl, version) 24, 385-392... 

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