Modeling isothermal

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

The adsorption of a number of organic pollutants on various solid surfaces was found to fit the Langmuir-model isotherm [139,143-145]. 

Waste type Solid phase type Model isotherm Y axis Intercept Slope X axis R2... 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

Figure 1. Adsorption isotherms for the pure calcined lamellar phase and MCM-41. The adsorption isotherm for the latter is compared with the model isotherm obtained using Eq. 6.

The BET (Eq. A2.3.4) or GAB (Eq. A2.3.5) equations relating water content of the sample and aw can be used to model isotherms. The constants in BET equations have to be determined for each temperature while GAB equation can accommodate the temperature effect as well (see Anticipated Results). Any commercial software (e.g., SAS, Statistica, S-plus, Minitab) with nonlinear regression feature can be used to determine constants in the BET or GAB equations. 

For an evaluation of the local model isotherm 6(p,T,Q) with constant interaction energy Q, the effects of multi-layer adsorption and lateral interactions between neighboring adsorbed molecules are considered by applying two modifications to the Langmuir isotherm (i) a multi-layer correction according to the well known BET-concept and (ii) a correction due to lateral interactions with neighboring gas molecules introduced by Fowler and Guggenheim (FG) [105] ... 

There is a growing interest in the presentation of physisorption isotherms in a generalized integral form. This approach was first applied to physisorption in the submonolayer region (Adamson et al., 1961), but much of the current interest is centred on the analysis of micropore filling isotherms. An apparent advantage is that it provides a means of constructing a series of model isotherms by systematically... 

It is of interest to compare the values of pore diameter obtained by molecular simulation and by the use of the corrected Kelvin equation. By comparing the nitrogen isotherm in Figure 12.6 with molecular simulation model isotherms, Maddox et al. (1997) have arrived at pore diameters of 4.1-4.3 nm. As indicated in Table 12.4, the corrected Kelvin diameters are 3.3-4.3 nm. The corresponding surface areas are 631 and 655 m2 g 1. In view of the assumptions in the model and the shortcomings of the Kelvin and BET equations, this level of agreement must be considered to be encouraging. 

Foubert, 1. (2003). Modelling isothermal cocoa butter crystallization Influence of temperature and chemical composition. University of Ghent, PhD, p. 263. 

Figure 1.12. Six ways of plotting the same data. Model isotherm according to Langmuir. The values of the Langmuir constant (in m N ) are indicated. Explanation in the text.

The Horvath-Kawazoe (HK) method is capable of generating model isotherms more efficiently than either molecular simulation (MS) or density functional theory (DFT) to characterize the pore size distribution (PSD) of microporous solids. A two-stage HK method is introduced that accounts for monolayer adsorption in mesopores prior to capillary condensation. PSD analysis results from the original and two-stage HK models are evaluated. 

The two-stage HK model isotherms are of the general form illustrated in Figure 1. The transition pressures and adsorbed fluid densities are calculated using the following procedure. 

Figure 4 Theoretical model isotherms for the original HK method (diamonds), the two-stage HK method (triangles), and DFT (squares) fitted to the experimental isotherm (circles) for nitrogen adsorption at 77 K on a granular activated carbon.

Textural Parameters. Adsorption-desorption isotherms of N2 at 77K were determined in a Micromeritics ASAP 2010 with a micropore system. Prior to measurement, the samples were outgassed at 140 C for at least 16 h. The specific surface area was determined by the BET method, assuming that the area of a nitrogen molecule is 0.162 nm [12]. Micropore volume was calculated by the t-plot method using the Harkins and Jura [13] thickness. We used model isotherms calculated from density functional theory (DFT) to determine the pore size distributions and cumulative pore volume of the pillared samples by taking the adsorption branch of the experimental nitrogen isotherm, assuming slit-like pores [14]. 

For faujasite-type zeolites and zeolite-A, where several molecules can occupy one cage, a special type of isotherm has been derived, the statistical model isotherm [49]. This isotherm treats each cage in the zeolite as a subsystem. Each cage can contain a fixed maximum number of molecules, and the molecules can interchange between the cages. Within a cage, interaction between the molecules is taken into account, which is not accounted for in the Langmuir description. 

The novel approach for calculation of pore size distributions, which is reported in the current study is based on recent developments in the materials science and in the theory of inhomogeneous fluids. First, an application of experimental adsorption data for well-characterized MCM-41 silicas enabled proper calibration of the pore size analysis. Second, an application of a modem theory to describe the behavior of inhomogeneous fluids in confined spaces, that is the non-local density functional theory [6], allowed the numerical calculation of model isotherms for various pore sizes. In addition, a practical numerical deconvolution method that provides a "best fit" solution representing the pore distribution of the sample was implemented [7, 8]. In this paper we describe a deconvolution method for estimating mesopore size distribution that explicitly allows for unfilled large pores, and a method for creating composite, or hybrid, models that incorporate both theoretical calculations and experimental observations. Moreover, we showed the applicability of the new approach in characterization of MCM-41 and related materials. 

In order to calculate the model isotherms, we first define a set of pore widths to be modeled and a set of pressure points at which to calculate quantity adsorbed. The set of pore widths can be chosen somewhat arbitrarily, but the pressure vector should be specifically constructed to properly weight all pore widths. The algorithm for calculating an isotherm point in the matrix of model isotherms, q(p,Hj), proceeds similarly for all the models considered here and is described in the following sections. 

The model isotherm for each pore size class was calculated by methods described previously [9], modified to account for cylindrical pore geometry. These calculations model the fluid behavior in the presence of a uniform wall potential. Since the silica surface of real materials is energetically heterogeneous, one must choose an effective wall potential for each pore size that will duplicate the critical pore condensation pressure, p, observed for that size. This relationship is shown in Figure 2. The Lennard-Jones fluid-fluid interaction parameters and Cn/kg were equal to 0.35746 nm and 93.7465 K, respectively. 

The matrix models were therefore calculated from a combination of the flat surface and model pore isotherms by the following algorithm Starting with the lowest pressure point, the amount adsorbed indicated by the flat surface model was compared to that of the pore model the flat surface isotherm was followed until the amount predicted by the pore model was the greater, then the pore model isotherm was followed for the remainder of the pressure vector. 

In this equation, z is the normal distance from the adsorbent surface, p is the particle density, U(z) the fluid - fluid potential and V(z) the wall potential, all at position z. Eq. 6 therefore defines the total integral heat of adsorption at any pressure point on a model isotherm calculated by DFT. 

Distributed Reactivity Model. Isotherm relationships observed for natural systems may well be expected to reflect composite sorption behavior resulting from a series of different local isotherms, including linear and nonlinear adsorption reactions. For example, an observed near-linear isotherm might result from a series of linear and near-linear local sorption isotherms on m different components of soft soil organic matter and p different mineral matter surfaces. The resulting series of sorption reactions, because they are nearly linear, can be approximated in terms of a bulk linear partition coefficient, KDr that is... 

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