Statistical Thermodynamic Model

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. 

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. 

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 rich metric structure of macroscopic thermodynamics also presents unusually stringent tests of theoretical models. Attempts to understand thermodynamic phenomena at a molecular level seem to demand improved dynamical and quantum statistical thermodynamic models that adequately incorporate the subtleties of quantum-mechanical valency and bonding interactions. Development of such models is an active area of modem physical chemistry research, but a more complete survey of the current molecular theory of gases and liquids is beyond the scope of the present work. 

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. 

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. 

Helfand (25,26,27,28,29) has formulated a statistical thermodynamic model of the microphases similar to that of Meier. This treatment, however, requires no adjustable parameters. Using the so-called mean-field-theory approach, the necessary statistics of the molecules are embodied in the solutions of modified diffusion equations. The constraint at the boundary was achieved by a narrow interface approximation which is accomplished mathematically by applying reflection boundary conditions. 

Above 1123° C. the U02+a.—which is not unsatisfactorily interpreted directly in terms of a simple statistical thermodynamic model—replaces the partially ordered U409 1/ phase over the whole composition range up to about U02>26 at 1400° C. It does not appear that a very large heat effect can be ascribed to the partial order complete disorder process relating to these two phases. 

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. 

Consider a non-reactive system consisting of a binary liquid alloy A-B and an oxide substrate such as AI1O3 at constant temperature. A simple statistical thermodynamic model has been developed (Li et al. 1989) to predict the contact angle and the work of adhesion isotherms, 0(XB) and Wa(XB), from the known values of contact angles... 

The development of the most important macroscopic and statistical thermodynamic models that describe adsorption phenomena on electrodes... 

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... 

This argument has descended onto the equation of state as a principal determinant of peculiar temperature dependences of hydrophobic effects. The statistical thermodynamic model discussed above, however, started with probabilities and fluctuations. But equations of state and fluctuations are connected by the most basic of results of Gibbsian statistical mechanics, e.g. Eq. (2.24), p. 27. Ad hoc models, such as the more simplistic lattice gas models, can be adjusted to agree with solubility at a thermodynamic state point, but if they don t agree with the equation of state of liquid water more broadly they can t be expected to describe molecular fluctuations of liquid water consistently and realistically. Thus, models of that sort are unlikely to be consistent with the picture explored here. 

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 ... 

Assume now that we are in the position to be able to calculate reliable pair potentials. The next task is to employ some statistical thermodynamic model which would permit us to pass from the pair complexes solvent-solute and solvent-solvent to a real liquid, A theoretical analysis of this problem is beyond the scope of this book, so that we restrict ourselves to stating that, in the conjucticn with ab initio calculations, the most sophisticated approach appears to be the statistical mechanics computer simulation of the finite system of N molecules in the volume V at the temperature T. The essence of the... 

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 Advance The Statistical Thermodynamic Model... 

SF theory is a statistical thermodynamic model in which chain conformations are formulated as step-weighted random walks in an interfacial lattice (Figure 2). A simple case involves the adsorption of a flexible, linear, homo-disperse, uncharged molecule at a uniform planar surface. Interactions among... 

Simple statistical thermodynamic models of adsorption [14] suggest that the equilibrium isotherm should be written as the ratio of two polynomials of the... 

In summary, diffraction techniques provide a powerful means of investigating the structure of electrolyte solutions. They give information about the pair correlation functions which can be directly related to modern theoretical techniques such as molecular dynamics calculations. This information can also be used to improve the statistical thermodynamic models of electrolyte solutions discussed in chapter 3. 

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