Statistical thermodynamic model for

Kerrick D. M. and Darken L. S. (1975). Statistical thermodynamic models for ideal oxide and silicate solid solutions, with applications to plagioclase. Geochim. Cosmochim. Acta, 39 1431-1442. 

Method of Cao, Knudsen, Fredenslund, and Rasmussen The method of Cao et al. [23] is based on a statistical thermodynamic model for pure liquids and liquid mixtures. It requires the input of the compound properties VM and AHv and two... 

The Center for Hydrate Research has been conducting hydrate experiments for over 30 years in efforts to improve flow assurance strategies. The first statistical thermodynamic model for hydrates was developed in 1959 and involved many assumptions, including the assumption that volume is constant. This model predicted hydrate formation temperatures and pressures reasonably well at temperatures near the ice point and at low pressures. However, as industry moves to deeper waters, there is a need for a hydrate model that can predict hydrate formation at higher temperatures and pressures. 

With the exception of the clathrate framework model, all these hypotheses appear to be qualitatively consistent with the available X-ray diffraction data on liquid water. It is argued by some investigators, however, that there are still significant inconsistencies between the most sophisticated statistical thermodynamic models for liquid water and the most sophisticated X-ray and neutron diffraction measurements [734-736]. The interpretation of these data from different experiments, using the concept of pair-correlation functions, shows discrepancies that are considered significant in terms of the instrumental precision, and the definitive answer seems not yet available [737J. 

The ffee-volume model has been used to describe the rapid decrease in solute diffusivity through human red cell membranes with increases in molecular size [96]. A statistical thermodynamic model for size distribution can then be employed to show that the diffusion coefficient of a solute is related to its MV ... 

The modem era of hydrate research is marked by the industrial adoption of the van der Waals and Platteeuw statistical thermodynamics model for the hydrate phase. The spectroscopic measurement of the hydrate phase, abetted with molecular simulation, led to accuracy improvements, and industrial applications to energy, seafloor stability, and climate change. With the above historical advances, consider modem thermodynamics of the hydrate phase itself. 

The most important test of our bimolecular potential is its use in statistical thermodynamic models for the prediction of phase equilibria. Monovariant, three-phase pressure-temperature measurements and invariant point determinations of various gas hydrates are available and are typically used to fit parameters in molecular computations. The key component needed for phase equilibrium calculations is a model of the intermolecular potential between guest and host molecules for use in the configurational integral. Lennard-Jones and Kihara potentials are usually selected to fit the experimental dissociation pressure-temperature data using the LJD approximation (2,4,6), Although this approach is able to reproduce the experimental data well, the fitted parameters do not have any physical connection to the properties of the molecules involved. 

The statistical thermodynamic model for hydrate equilibria developed by van der Waals and Platteeuw in 1959 (2) was generalized by Parrish and Prausnitz for prediction of hydrate dissociation pressures (42). The method of... 

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

If stationary phase interactions are negligible the lattice statistical thermodynamic model and the solvophobic model predict similar results. The strength of the lattice statistical thermodynamic model is that it can explain the shape selectivity observed for certain stationary phases and can accommodate silanophllic interactions. 

A central issue in statistical thermodynamic modelling is to solve the best model possible for a system with many interacting molecules. If it is essential to include all excluded-volume correlations, i.e. to account for all the possible ways that the molecules in the system instantaneously interact with each other, it is necessary to do computer simulations as discussed above, because there are no exact (analytical) solutions to the many-body problems. The only analytical models that can be solved are of the mean-field type. 

A substantial number of models have been proposed for the insect cells/BEVS system ranging from empirical approaches [103], through unstructured [102, 104],structuxed and mechanistic [60,61] to a recent kinetic and statistical-thermodynamic model [105]. 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

For accurate prediction of the hydrate formation conditions for acid gases, the advanced methods are preferred. The more advanced methods for predicting hydrate formation are based on the work of van der Waals and Platteeuuw (1959). This is a statistical thermodynamic model. 

From the point of view of molecular theory, the coefficients dpu jdp2 are fundamentally related to structural properties of these fluids - OZ direct correlation functions (Eq. (6.71), p. 141) - as is discussed in detail subsequently in Section 6.3 on the Kirkwood-Buff theory. Alternatively, these coefficients could be explicitly evaluated if an explicit statistical thermodynamic model, as in the discussion here, were available for the unmixed fluids. Finally, these comments indicate that much of the iirformation supplied by these coefficients is susceptible to measurement... 

The same approach can be applied to investigate the explosivity conditions of the H20-NaCl system. We have selected the Anderko-Pitzer (AP) equation of state,which is based on realistic physical hypotheses. It describes H20-NaCl by means of statistical thermodynamic models developed for dipolar hard spheres. This assumption is reasonable at high temperatures, where NaCl is known to form dipolar ion pairs. However, for this reason, this equation of state is only applicable above 573 K, 300°C. 

In this work, a gas chromatographic system has been used to obtain binary sorption equilibria for nitrogen-oxygen, methane-ethane and propane-cyclopropane mixtures on 5A molecular sieve. The experimental measurements have also been compared with the predictions from the statistical thermodynamic model of Ruthven (11). The model is capable of predicting binary sorption equilibria from the parameters derived from the pure-component isotherm measurements. 

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