Adsorption Integral Equation Research Articles

Energetic and structural heterogeneity of activated carbons determined using dubinin isotherms and an adsorption potential in model micropores

Adsorption on microporous carbons is investigated in terms of their micropore structure and their adsorption energy heterogeneity. The pore size heterogeneity is proposed to be a controlling factor for the adsorption energy heterogeneity. The relationship between adsorption energy and pore sizes is based on the dependence of the molecule-pore interaction potential well depth as a function of the pore width. The adsorption integral equation is used to analyze data generated from the DR and DA isotherms using typical experimental conditions. It is observed that the adsorption systems described by these isotherms are strongly heterogeneous from the standpoint of their adsorption energetics which are characterized by asymmetrical energy distributions with a broadening toward high energies. It is also shown that these distributions correspond to relatively narrow and symmetrical peaks characterizing the distribution of pore sizes. The strong asymmetrical relationship between adsorption energy and pore width magnifies the energetic heterogeneity even though pore size distributions show only modest dispersion. It is demonstrated that the Henry's law region appears as a natural consequence of the assumed model.

Comparison of semianalytical methods to analyze complexation with heterogeneous ligands

The binding of ions to natural ligands is influenced by the chemical heterogeneity of these ligands. Two approaches to calculate the affinity distribution are compared in terms of theory and practicality. The local isotherm approximation (LIA) methods use an approximation of the local binding function in order to solve the integral adsorption equation for the distribution function. The differential equilibrium function (DEF) method is based on the law of mass action, but it is shown that the method can also be interpreted within the LIA concept

Local exact and approximate solutions of the adsorption integral equation with a kernel of a Langmuir-like isotherm: Determination of adsorption energy distribution

The local exact and approximate solutions of the adsorption integral equation are presented. The Langmuir-like adsorption isotherm is assumed as a kernel. The exact local solution is compared with Sips' method. The accuracy and resolution of truncated approximations are discussed. The influence of experimental temperature on calculated results is also considered. Lateral interactions are taken into account for the random topography model of the adsorption sites.

Adsorption equilibria of pure vapors and their binary mixtures on activated carbon Part I. Single-component equilibria

Absorption and desorption equilibria of three organic vapors on two kinds of activated carbon are investigated theoretically and experimentally. The application of the adsorption integral equation to microporous adsorbents is discussed. It is shown that the Dubinin theory of volume filling of micropores (TVFM) provides an improved method for describing the adsorption of organic vapors in the micropores of the carbon. From adsorption and desorption experiments, the transition from micropore filling to capillary condensation in mesopores can be observed. The corresponding adsorption potentials are related to a pore radius of about 1.6 nm. A modified Dubinin—Astakhov equation is proposed, which takes into account the end of micropore filling and the characteristic curves by means of a limiting potential A Gr. This equation describes the isothermal field of all six investigated systems very well in a wide temperature range (20 – 150 °C) with temperature-invariant parameters.

Characterization of silicas by inverse gas chromatography at finite concentration: Determination of the adsorption energy distribution function

Amorphous and crystalline silicas, eventually modified by heat treatment or by surface grafting of alkyl chains, were investigated by inverse gas chromatography, a method which easily provides adsorption isotherms of various linear alkanes. The isotherms were analyzed in order to obtain the adsorption energy distribution of alkane probes. The distributions were calculated using a local solution of the adsorption integral equation which is evaluated and discussed. Silicas show typical energy distribution patterns. Furthermore, the grafting of hexadecanol chains on the silica surface brings the greatest changes in the energy distributions.

Determination of adsorption affinity distributions: A general framework for methods related to local isotherm approximations

A family of methods is presented for the determination of the adsorption affinity distribution function for a heterogeneous surface from single component adsorption data. It is possible to deal with different types of local isotherms and with random and patchwise heterogeneity. The general concept is that an approximation of the local isotherm is used to solve the integral adsorption equation analytically for the distribution function without making a priori assumptions about the distribution. The method is worked out for FFG type equations as local isotherm with an interaction parameter incorporated. Examples are given for the Langmuir local isotherm. The simplest member of this family of local isotherm approximations (LIA) is the step function (STEP), known as the condensation approximation (CA). The first order affinity spectrum (AS 1) strongly resembles the CA distribution. Both methods result in general in a too wide affinity distribution. An alternative is the use of a linear approximation (LINA) of the local isotherm. These LINA methods cause a widening and an asymmetric deformation of the true distribution. The asymptotically correct condensation approximation (ACCA) is a member of this group. A substantially better approximation is achieved by considering the local isotherm and its approximations on a logarithmic concentration (or mole fraction) scale (LOGA). The distributions obtained with the Rudzinski Jagiello (RJ) method and the second order affinity spectrum (AS 2) method can be interpreted as members of the LOGA group. Parameter optimization in the LOGA case has resulted in two other solutions which are better approximations than the RJ and AS 2 method.

A new numerical method for calculating the energy distribution from adsorption isotherm data

A new numerical method for solving the integral equation of adsorption with respect to the energy distribution function is proposed on the basis of the Stieltjes integral transformation and by utilizing smoothing cubic spline functions. This method is tested by using the model adsorption isotherms relating to the distributions consisting of three and five Gaussian peaks. For the purpose of illustration the energy distribution is calculated according to this method by using low-temperature nitrogen adsorption data on Aerosil.

Physical adsorption of gases on heterogeneous surfaces. Model study of the effects of simultaneous vertical and lateral interactions by monte carlo methods

A model study has been undertaken to elucidate the interplay between vertical and lateral interaction effects in the physical adsorption of gases on heterogeneous solid surfaces. The surface model employed is a square lattice with a random distribution of adsorption sites of different energies on which adsorbed molecules interact only with their nearest neighbours. Isotherms are generated by simulating the system in the grand canonical ensemble by Monte Carlo methods. These isotherms are subsequently subjected to analysis by the adsorption integral equation using various local isotherm models, in an attempt to recover the adsorption energy distributions. When lateral interactions are ignored, the analysis yields an adsorption energy distribution which (a) is centred on a mean value which is too high and (b) has a ‘spread’ which is too small. These effects are correlated with the magnitude of the neglected lateral interactions and a possible physical interpretation of the mean energy calculated by such means is given. Differential heats of adsorption as well as isotherm ‘data’ are presented for the model system considered.

Calculation of the adsorption energy distribution from the adsorption isotherm by singular value decomposition

An algorithm has been developed to obtain the energy distribution function from an adsorption isotherm. The ill-posedness of the integral adsorption equation is taken into account and negative values in the calculated distribution function are excluded. Computations are performed with computer program CAESAR (computed adsorption energies, SVD analysis result). The method is tested with simulated isotherm data and the results are very promising. The energy distribution functions of four carbon-black samples confirm that graphitization at high temperatures leads to nearly homogeneous surfaces. An analysis of different Aerosils shows that chemical modifications of the surface are reflected in the energy distribution. Comparison of the results obtained with CAESAR with those of other methods indicates that the same trends are observed.

Numerical solutions of the adsorption integral equation utilizing the spline functions

Detailed studies dealing with numerical solutions of the fundamental integral equation representing the overall adsorption isotherms are carried out utilizing different types of spline function. The successive use of the smoothing cubic spline functions is recommended to calculate the energy distribution function from the adsorption isotherm. A new computer program EDCAIS (energy distribution computation from adsorption isotherm utilizing the smoothing spline functions) is developed for quasi-Langmuir local isotherms involving multilayer correction. EDCAIS is tested using the theoretical adsorption isotherms relating to five types of energy distribution: constant distributions and distributions composed of one, three and five gaussian peaks.

Exact solution of the adsorption integral equation for a heterogeneous surface

An exact method of solution of the adsorption integral equation using finite and realistic limits of the heats of adsorption and which is applicable to any analytic site-energy distribution is given. The optical feature is the postulation and use of a set of transformations to reduce the finite-limit problem to one of limits (0, ∞). The transformed integral equation is then inverted using the theory of Stieltjes transforms. The method thus avoids the limitations inherent in earlier work which employed restrictive assumptions about the heats of adsorption.

Adsorption on heterogeneous surfaces: Consequence of assuming the Jovanović equation for local adsorption

The consequences of assuming Jovanovic local adsorption isotherm are discussed on the basis of integral adsorption equation for heterogeneous surfaces. Analytical solutions are presented for Dubinin-Radushkevich adsorption isotherm.

The adsorption integral equation: A new method of solution

A classification of the adsorption integral equation in to finite and infinite limit problems is made. A method of solution based on the theory of finite Hilbert transforms is proposed.

Investigation of solutions of the integral adsorption equation in respect to the range of integration of the adsorption energy

A new method is proposed for solving the integral equation for the overall adsorption isotherm, with respect to the energy distribution function. This method can be applied for arbitrary integration limits; finite or infinite. The purpose of the calculations carried out was to examine the solution of the integral equation in relation to accepted integration limits.