Adsorption molecular-statistical calculation

Molecular Statistical Calculation of Adsorption Equilibria of Vapors on Xeolites... 

Table II on p. 54 gives some results of our molecular statistical calculations of the heat of nonspecific adsorption of simple molecules in the cavities of type A zeolites. Molecular statistical calculation of thermodynamic characteristics of adsorption in the cavities of type X zeolites are more difficult because of the more complicated structure of the lattice.

The studies presented here are based essentially on the principle of comparison of the thermodynamic characteristics of adsorption measured on a number of organoderivatives of layer silicates and silica, with the data obtained from the molecular statistic calculations involving the portions of surface, which model the real surface of the materials studied. These surface portions were chosen on the basis of the adsorbent structure analysis and complex physico-chemical studies of the modifying layers structure. 

Theoretical molecular statistic calculations of adsorption thermodynamic properties for colloid systems are generally believed to be a quite cumbersome problem. This is mainly... 

Fundamental research into adsorption and chemical modification has helped to elucidate chromatographic processes. This research has led to new developments in gas adsorption chromatography by A. Kiselev, Yashin (Experimental Design Office, Dzerzhinsk), Poshkus (Institute of Chemistry and Chemical Technology, the Lithuanian Academy of Sciences, Vilnius) and co-workers (2, 17, 288-290). Together with theoretical investigations and molecular statistical calculations, this method can be used to determine the structural parameters of the adsorbate molecule (this direction of research is referred to as chromatoscopy). 

A. G. Bezus, A. V. Kiselev, A. A. Lopatkin, and P. Q. Du,/. Chem. Soc., Faraday Trans. 1, 74, 367 (1978). Molecular Statistical Calculation of the Thermodynamic Adsorption Characteristics of Zeolites Using the Atom-Atom Approximation. 1. Adsorption of Methane hy Zeolite NaX. 

It is very important to obtain accurately the first 2 constants (Ci and C2 or Ki and K ) from the experimental data in order to compare them with those calculated from the molecular-statistical theory. Here, Cl and C2 constants were determined from the experimental isotherm by the least squares method with the aid of a computer. However, for practical purposes, it is necessary to describe the isotherms to a higher adsorption level. From Figures 3 and 6, it can be seen that in order to reproduce the isotherm up to — 75% saturation of the adsorbent and to calculate a at different values of p and T, it is sufficient to determine only 2 more constants, C3 and C4. 

It is well known that any theory of molecular adsorption which has a molecular statistical basis must lead to the Henry equation in the limiting case of small concentrations. The Polanyi theory and its modifications do not satisfy this condition. In contrast to this, the theories of adsorption which take into consideration the adsorbate-adsorbent and adsorbate-adsorbate interactions on the molecular level give the limiting Henry equation. Therefore, we can use these theories to calculate the Henry constant, the constant of adsorbate-adsorbent equilibrium. I have shown... 

Pore size distribution has been calculated using DFT method for all materials. Indeed, DFT, based on molecular statistical approach, is applied over the complete range of the isotherm and is not restricted to a confined range of relative pressure or pore sizes. Pore size distribution is calculated by fitting the theoretical set of adsorption isotherms, evaluated for different pore sizes, to the experimental results. For instance Figure I shows the N2 experimental isotherm together with the DFT best fitting isotherm for MSA sample. 

Instead of the classical approaches, a molecular-based statistical thermodynamic theory can be applied to allow a model of adsorption to be related to the microscopic properties of the system in terms of fluid-fluid and fluid-solid interactions, pore size, pore geometry and temperature. Using such theories the whole range of pore sizes measured can be calculated using a single approach. Two simulation... 

Hence, the temperature coefficient of k1 having been measured, for an absolute calculation of k only kt) and bo must be known, and not the heat of adsorption, X. At the moment we are concerned with b0. A simple statistical estimate can be based on the assumption that in the absence of adsorption energy the adsorption space is filled at a proportion given by the ratio of the molecular adsorption volume (liquid volume Fm) to the molecular gas volume... 

In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

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. 

As expected, the total interaction energies depend strongly on the van der Waals radii (of both sorbate and sorbent atoms) and the surface densities. This is true for both HK type models (Saito and Foley, 1991 Cheng and Yang, 1994) and more detailed statistical thermodynamics (or molecular simulation) approaches (such as Monte Carlo and density functional theory). Knowing the interaction potential, molecular simulation techniques enable the calculation of adsorption isotherms (see, for example, Razmus and Hall, (1991) and Cracknell etal. (1995)). 

Tabulated data for experimental adsorption isotherms are fitted with analytical equations for the calculation of thermodynamic properties by integration or differentiation. These thermodynaunic properties expressed as a function of temperature, pressure, and composition are input to process simulators of atdsorption columns. In addition, anaJytical equations for isotherms are useful for interpolation and cautious extrapolation. Obviously, it is desirable that the Isotherm equations agree with experiment within the estimated experimental error. The same points apply to theoretical isotherms obtained by molecular simulation, with the requirement that the analytical equations should fit the isotherms within the estimated statistical error of the molecular simulation. 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

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