Gaussian distribution of adsorptive

The interpretation of adsorption isotherms in terms of a surface displaying a symmetrical (Gaussian) distribution of adsorptive potentials to the adsorbate has been successfully applied to a number of systems but marked exceptions have been found, particularly with crystals that were selected as likely to have completely homotattic surfaces. These exceptional adsorption isotherms are readily interpreted, however, as the result of a sum of two, or occasionally three, distinctly different Gaussian distributions, presumably deriving from the same number of surface constituents. Among these constituents the expected homotattic substrate, which is associated with a particular crystal face, can always be identified. 

A previous paper of this series (13) describes the analysis of adsorption iso- therms for heterogeneous surfaces in terms of a Gaussian distribution of adsorptive potentials. The distribution of adsorptive potential energies may not, however, have a form that is symmetrical about a mean it could possess enough asymmetry so that the adsorption isotherm could not be described by a model that assumed a Gaussian distribution. 

The first such solutions were carried out by Ross and Olivier [1, p. 129 6,7]. Using Gaussian distributions of adsorptive potential of varying width, they computed tables of model isotherms using kernel functions based on the Hill-de Boer equation for a mobile, nonideal two-dimensional gas and on the Fowler-Guggenheim equation [Eq. (14)] for localized adsorption with lateral interaction. The fact that these functions are implicit for quantity adsorbed was no longer a problem since they could be solved iteratively in the numerical integration. 

Gaussian distribution of adsorption over the surface sites i.e. he assumed VIVm. = exp(-/fc 3). 

Parameter k of Equation (4.10) is an expression of the breadth of the Gaussian distribution of the cumulative micropore volume IF over the normalized work of adsorption sfifi, and is therefore determined by the pore structure. Thus B also (cf. Equation (4.13)) is characteristic of the pore structure of the adsorbent, and has accordingly been termed the structural constant of the adsorbent. ... 

FIGURE 2.1 The linear isotherm of adsorption and the corresponding Gaussian distribution of the analyte s concentration in the chromatographic band. 

The distribution of adsorptive potentials of the adsorbent surface is again taken as the Gaussian probability function ... 

Three commercial activated carbons were used BPL, CAL and GAe, manufactured by Chemviron, Calgon and CECA respectively. In addition, sample GAe-oxl was prepared by oxidation of GAe in aqueous solution of (NH4)2S20g and further pyrolysis in N2 flow at 773 K [5]. The specific surface areas were obtained applying the BET and Dubinin-Asthakov equations to the adsorption of N2 at 77 K and CO2 at 273 K respectively. Moreover, the C02 adsorption data permitted the evaluation of the micropore size distributions and the mean value of pore width using the Dubinin-Stoeckli equation [6] which supposes a gaussian distribution of pore sizes. 

Figure 3 Comparison of PSDs obtained using the Dubinin-Stoeckli (DS), Horvalh-Kawazoe (HK), and density fitnctional theory (DFT) methods to interpret an isotherm generated from molecular simulation of nitrogen adsorption in a model carbon that has an Gaussian distribution of slit pore widths (18]. Results are shown for mean pore widths of 8.9 A (left) and 16.9 A (right).

Distribution of adsorption centers on the surface of a non-modified silica gel is rather exponential, while in the case of carbon-silica adsorbents it is double or triple gaussian. Correlations were found between the topography and morphology of carbon deposits and the adsorption properties of carbosils (magnitudes of q°(, shape of the function X(E) and TEM photographs)[15-17j. Topography of the carbosils is of an evident patch-wise type, which is indicated by the distinct peaks in the X(E) curves. It also follows from the fact that the global function X(E) for such surfaces may be expressed as [29 ... 

This section will address the DR local isotherm and the Gaussian distribution of the form given in eq. (4.4-5). The volume of the micropore occupied by the adsorbate at a given adsorption potential A is ... 

This method was successful in the first attempts to solve equation (15) using a local isotherm equation that allowed adsorbate lateral interactions within patches. The ingenious graphical method, now referred to as the Ross and Olivier method of analysis, has been described in their monograph and in a condensed account by Ross. The technique used by Ross and Olivier employs a Gaussian probability function for the distribution of adsorption energies, i.e. a two-parameter generalized distribution of the form ... 

There is no unique structure within an activated carbon which provides a specific isotherm, for example the adsorption of benzene at 273 K. The isotherm is a description of the distribution of adsorption potentials throughout the carbon, this distribution following a normal or Gaussian distribution. If a structure is therefore devised which permits a continuous distribution of adsorption potentials, and this model predicts an experimental adsorption isotherm, this then is no guarantee that the stmcture of the model is correct. The wider experience of the carbon scientist, who relates the model to preparation methods and physical and chemical properties of the carbon, has to pronounce on the reality or acceptance value of the model. Unfortunately, the modeler appears not to consult the carbon chemist too much, and it is left to the carbon chemist to explain the limited acceptability of the adopted stractures of the modeler. 

Such deviations occur when distributions of adsorption site energies do not fit a Gaussian-type (or related distribution function). Then, the obtained experimental isotherm will not be linearized by the conventional Langmuir, BET and DR adsorption equations. If the continuity of the distribution curve is disturbed in some way (e.g. by selective oxidation to widen some parts of the porosity during an activation process) then deviations will occur from the model equations. Elaborations of equations to obtain a better fit are mathematical devices to correct for deviations to the distribution curves but do little to explain the causes. 

Figure 4.11. The uniqueness of the adsorption isotherm. If the distribution of adsorption site potential energies does not fit a Normal-Gaussian-type distribution function, as above in the figure, then the obtained experimental isotherm is not linearized by the Langmuir,...

In the case of Gaussian and uniform distributions of the adsorption energy, the smearing of the phase transition region in the the first as well as higher layers was observed. Thus, insead of vertical jumps, the adsorption isotherms exhibited only finite slope even at quite low temperatures. This result is consistent with the predictions of Dash and Puff [32]. 

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