An overall adsorption quantity for a heterogeneous solid is usually expressed by an integral equation, which contains a distribution function that describes heterogeneous properties of this solid. The calculation of this distribution function is an ill-posed problem. The current article shows that the difficulties arising from the ill-posed nature of an adsorption equation can be overcome with the regularization method. This work presents general principles of regularization for solving the ill-posed problems without detailed mathematical considerations. The application of the regularization method to calculate a distribution function from any overall adsorption functions is illustrated with both simulated and experimental adsorption isotherms.

Adsorption equilibria of argon, nitrogen, and methane on the 13X molecular sieve and the AS activated carbon were measured at five temperatures over a wide pressure range from 0.1 to 20 MPa using a microbalance. From experimental adsorption isotherms, which are excess functions, the absolute isotherms were calculated using equations of state for a real gas phase. Grand canonical Monte Carlo simulations and density functional theory calculations were carried out in order to explain specific features of the resulting isotherms. Different thermodynamic functions evaluated from the excess and absolute adsorption isotherms were analyzed over a wide pressure range at the average temperature of the range studied.

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Systematic studies of the energetic heterogeneity of graphitized and ungraphitized reference carbonaceous materials have been carried out using several standard adsorption isotherms available in the literature. Adsorption energy distribution functions have been calculated by using a new algorithm based on a regularization method. Analysis of the calculated energy distributions for various reference carbons provides important comparative information about their energetic heterogeneity. It has been shown that the surface properties of the reference materials vary depending on differences in the degree of their graphitization.

Various problems associated with numerical evaluation of the adsorption energy distribution function, F(Uii), from the experimental excess adsorption isotherms of binary nonelectrolytic liquid mixtures on heterogenous solid surfaces have been discussed. Several methods known in the literature did not address the numerical ill-posed character of the integral equation, which has been used for calculating F(f/21). In the current work two alternative approaches, which take into account the ill-posed nature of this integral equation, have been presented. In the first approach a regularization method has been employed, whereas in the second one an expansion of the Stieltjes' transformation has been utilized. Also, the problem of choice of the local adsorption model has been considered. A representation of the overall excess adsorption isotherm in the form of the Aft-plot has been found to be useful for selecting this model. Illustrative calculations have been carried out for various mixtures on the same adsorbent and for one mixture on different adsorbents. Such as methodological approach is especially useful for analyzing differences between the systems studied. By use of the regularization method, the distribution functions F(t/21) for n-hexane/ 1-hexanol, n-hexane/toluene, and n-hexanol/ toluene on an active carbon have been calculated and compared.Another comparison of the IWn) functions have been presented for water/ethanol on ZSM-5 zeolites of different Si/Al ratios.