Localized adsorption isotherm

The local adsorption isotherm 0L is represented by the original D-A equation and /P(x) is the micropore size distribution ranging from xmm to xmilx (the lower and upper limits of the slit-like... 

In this method, the local adsorption isotherms are obtained from the DR equation (Equation 4.16), and the relation between the characteristic energy (E0) and the mean pore width (II) is obtained from the equation proposed by Stoeckli et al. [39] when E0 is between 42 and 20kJ/mol... 

Let 0(P,T) be the observed adsorption function, usually obtained as 0(F) — i.e., as an adsorption isotherm. It is assumed that the adsorbent surface can be treated as consisting of noninteracting regions, each of which can be considered as homogeneous in nature and obeying a local adsorption isotherm, 0(P), or, in general, an adsorption function. 0(P>T,Q), where Q denotes the adsorbent... 

For each pressure, the assumed local adsorption isotherm function allows 0 to be determined as a function Q, while from a simple correspondence principle, the experimental adsorption isotherm 0 (P) allows determination or a first approximation of F as a function of Q. From the plots of F vs. Q and 0 vs. Q, a subsidiary plot is made for each pressure of 0 vs. the corresponding F value. The area under this plot then gives a calculated 0 for that pressure. The procedure is then repeated for other assumed pressures. The 0ealcd. values will in general differ from the experimental ones for the various pressures, and for each, a second approximation to F is made using the relationship ... 

In this paper we present a new characterisation method for porous carbonaceous materials. It is based on a theoretical treatment of adsorption isotherms measured in wide temperature (303 to 383 K) and pressure ranges (0 to 10000 kPa) and for different adsorbates (N2, CH4, Ar, C3H8 and n-C4Hio). The theoretical treatment relies on the Integral Adsorption Equation concept. We developed a local adsorption isotherm model based on the extension of the Redlich-Kwong equation of state to surface phenomena and we improved it to take into account the multilayer formation. The pore size distribution fimction is assumed to be a bi-modal gaussian. By a minimisation procedure, it is possible to determine the bi-modal pore size distribution function witch can be used for purely characterisation purposes or to predict adsorption isotherms. 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

Local adsorption energies, e, local adsorption isotherms, dip, T, e), local monolayer capacities, c ax, and adsorption energy distribution functions, /(e), for... 

Gavril, D. An inverse gas chromatographic tool for the experimental measurement of local adsorption isotherms. Instnim. Sci. Technolog. 2002, 30 (4), 397-413. 

Others already employed local adsorption isotherms obtained from density functional theory in their calculations of pore size distributions for MCM-41 [16-17]. However, desorption data were used, which imposes two severe limitations on the results of calculations. 

For multicomponent systems, the local adsorption isotherm for a given micropore is assumed to follow the extended Langmuir equation. 

The prediction of multicomponent equilibria based on the information derived from the analysis of single component adsorption data is an important issue particularly in the domain of liquid chromatography. To solve the general adsorption isotherm, Equation (27.2), Quinones et al. [156] have proposed an extension of the Jovanovic-Freundlich isotherm for each component of the mixture as local adsorption isotherms. They tested the model with experimental data on the system 2-phenylethanol and 3-phenylpropanol mixtures adsorbed on silica. The experimental data was published elsewhere [157]. The local isotherm employed to solve Equation (27.2) includes lateral interactions, which means a step forward with respect to, that is, Langmuir equation. The results obtained account better for competitive data. One drawback of the model concerns the computational time needed to invert Equation (27.2) nevertheless the authors proposed a method to minimize it. The success of this model compared to other resides in that it takes into account the two main sources of nonideal behavior surface heterogeneity and adsorbate-adsorbate interactions. The authors pointed out that there is some degree of thermodynamic inconsistency in this and other models based on similar -assumptions. These inconsistencies could arise from the simplihcations included in their derivation and the main one is related to the monolayer capacity of each component [156]. 

The multiphase ideal adsorbed solution theory (MIAST) is another model of the family of adsorbed solutions. Corrtraiy to HAST, an energy distribution function is assumed with differences of the local or molectrlar site energy. Therefore, every site has its own local adsorption isotherm arrd the adsorbate concentration differs from site to site. The adsorbate is not considered as a homogeneous phase but as a multiphase system. 

Using DA as the local adsorption isotherm, and Eq = k/x, Jaroniec et al. (1988) assumed the following micropore size distribution ... 

The local adsorption isotherm takes the form of the Langmuir equation ... 

Other Local Adsorption Isotherms Energy Distribution... 

Recognising that the local adsorption isotherm is not Langmuir and the energy distribution is not uniform, the other forms of local isotherm as well as energy distribution can be used in eq. (6.3-1) or (6.3-2). Such a combination is infinite however we will list below a number of commonly used local adsorption isotherm equations and energy distributions. 

The local adsorption isotherm equations of the form Langmuir, Volmer, Fowler-Guggenheim and Hill-de Boer have been popularly used in the literature and are shown in the following Table 6.3-1. The first column shows the local adsorption equation in the case of patchwise topography, and the second column shows the corresponding equations in the case of random topography. Other form of the local isotherm can also be used, such as the Nitta equation presented in Chapter 2 allowing for the multisite adsorption. 

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