Integral adsorption equation

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]. 

The model is based the Integral Adsorption Equation (lAE) concept [14, 15, 16]. We have developed it for our specific experimental conditions, i.e., for the treatment of excess... 

Initially the first two possibilities of integral adsorption equation were intensively expolored by investigators who used almost exclusively the Langmuir adsorption isotherm [223-231]. Then, the type of the topography of adsorption sites is of free choice. The mathematical forms of the overall adsorption isotherms depend only on the shape of the energy distribution functions, which characterize... 

In 1970s and 1980s several numerical methods were proposed in order to find the distribution energy functions of adsorption on the basis of tabulated data of experimental adsorption isotherm. From a mathematical point of view the integral adsorption equation is the Fredholm integral equation of the first kind. The particular nature of this equation poses severe difficulties to its solution and strict limits to the range of numerical methods that can be used in such a task. 

In this paper, we have presented and tested a model which allows the calculation of adsorption isotherms for carbonaceous sorbents. The model is largely inspired of the characterization methods based on the Integration Adsorption Equation concept. The parameters which characterize the adsorbent structure are the same whatever the adsorbate. In comparison with the most powerful characterization methods, some reasonable hypothesis were made the pore walls of the adsorbent are assumed to be energetically homogenous the pores are supposed to be slit-like shaped and a simple Lennard-Jones model is used to describe the interactions between the adsorbate molecule and the pore wall the local model is obtained considering both the three-dimension gas phase and the two-dimension adsorbed phase (considered as monolayer) described by the R lich-Kwong equation of state the pore size distribution function is bimodal. All these hypotheses make the model simple to use for the calculation of equilibrium data in adsorption process simulation. Despites the announced simplifications, it was possible to represent in an efficient way adsorption isotherms of four different compounds at three different temperatures on a set of carbonaceous sorbents using a unique pore size distribution function per adsorbent. 

Determination of the equilibrium spreading pressure generally requires measurement and integration of the adsorption isotherm for the adhesive vapors on the adherend from zero coverage to saturation, in accord with the Gibbs adsorption equation [20] ... 

For orthokinetic conditions the ratio of adsorption halftime to coagulation halftime can be calculated by integrating the adsorption Equation (6) and the coagulation Equation (2). 

At high surfactant concentrations the surface pressure - area curves tend towards the surface pressures of the pure surfactant i(AII -r 0). Thus the integrals in equation 15 appear to be zero for > 1.4 nm2 molecule" and the adsorptions are then equal to the adsorptions for the monolayer-free system. In contrast, the Pethica equation at this area still imposes a significant correction factor on the adsorption the slope (3II/3 Inmg) for Am = 1.4 nm molecule" equals that for the monolayer-free system but (2m-2m)/... 

Suppose that the interaction forces establish an energy barrier that retards the motion of particles both toward and away from the collector. If this barrier reduces the adsorption and desorption rates significantly, particles near the primary minimum will have time to achieve a balance between the interaction forces and Brownian motion, before their population changes. Integration of Equation (6) with j 0 and D = mkT leads Lo a Boltzmann distribution... 

If we assume that the adsorption isotherm for the chemisorbate is of the Temkin form, then i k" cMcoh exp(-a/0ads), where a is the transfer coefficient, with a value usually close to 0.5, and / is the inhomogeneity factor of the Pt surface, with a value of 10-11. Integrating this equation assuming that cMeoH and E are constant, we obtain... 

The other method for the comparison of enthalpies depends upon the integration of Equation 5 as indicated by Equation 6 for each of a set of adsorption iso-... 

The use of this identity is equivalent to the integration of the Gibbs adsorption equation (1-4). 

Enthalpy Change. The enthalpy change measured by the heats of immersion (smooth curve) and calculated with Equation 7 (plotted points) is compared in Figure 6. The agreement is satisfactory, since the integration of the Gibbs adsorption equation depends so strongly upon the extrapolation of the adsorption isotherm to x = 0. 

Equation (2.34) is often referred to as the Gibbs adsorption equation where the interdependence of r and p is given by the adsorption isotherm. TTie Gibbs adsorption equation is a surface equation of state which indicates that, for any equilibrium pressure and temperature, the spreading pressure II is dependent on the surface excess concentration r. The value of spreading pressure, for any surface excess concentration, may be calculated from the adsorption isotherm drawn with the coordinates n/p and p, by integration between the initial state (n = 0, p = 0) and an equilibrium state represented by one point on the isotherm. 

As done in the case of Langmuir adsorption isotherm, integration of Equation (25) is performed with the condition that at f = 0 the substrate concentration in the liquid phase is that in equilibrium with the initial photoadsorbed amoimt, Cl,o this initial concentration is imknown, but it may be determined... 

The adsorption of CO occurs sequentially with Au CO as an intermediate product. As the carbon monoxide concentration in the trap is constant, all reaction steps are taken to be pseudo-first-order in the simulations. Purely consecutive reaction steps do not fit the experimental data, and it is therefore essential to introduce a final equilibrium (1.2). The fits of the integrated rate equations to the data are represented by the solid lines in Fig. 1.36b and are an excellent match to the experimental results. 

In these equations = (x,y, z,(p,, is the energy of interaction of the molecule with the adsorbent surface, the integration extends over the spatial variables (x,y) which specify the point on the adsorbent surface above which the molecule is situated, the variable z which represents the distance between the molecule and the adsorbent surface, and the Eulerian angles (y , describe the orientation of the molecule with respect to the surface A is the surface area over which (x,y) integration is performed. Calculation of the integrals in equation (1) allows the thermodynamic quantities for the adsorption of a quasi-rigid molecule of the species i to be expressed as follows ... 

However, in most cases of interest, the sixfold integration in (1) is incapable of numerical solution. Calculations performed so far for related models are those for the adsorption of the hydrocarbon molecules in zeolite cell voids, see, for example, [13]. In such cases, the spatial integration in equation (1) is restricted to a relatively small volume which is determined by the van der Waals radii of the atoms which constitute the molecule and... 

For a continuous heterogeneity the expression for the overall complexation of protons with all the different site types is a multiple integral equation. A relevant more simple, but still complicated situation, is that of e.g. proton and metal ion adsorption on oxides. For such a situation and adsorption from an indifferent background electrolyte the integral proton adsorption equation is ... 

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