Calculation of the BET-Compatible Surface Area from Any Type I Isotherms Measured above the Critical Temperature.
In our previous papers [Tóth, J., et al., J. Colloid Interface Sci. 212, 402 (1999); 212, 411 (1999)] it has been proven that a BET-compatible specific surface area, a(s)(c)(N(2), 77), can be calculated from any Type I monolayer isotherms and from the monolayer domain of Type II isotherms, both measured below the critical temperature. In this paper, the calculation of a(s)(c)(N(2), 77) from Type I isotherms at temperatures above the critical temperature is discussed. These calculations are also based on the thermodynamical consistent form of the Tóth (T) equation. This relationship can be derived from the differential equation dn(s)/n(s)=[psi(p)](-1)dp/p, where the function psi(p) is calculable from the measured isotherms and includes the change in the surface free energy. The basic problem of the calculation of a(s)(c)(N(2), 77) is the determination of the upper limit of integration (p(u)) required to solve this differential equation. It is demonstrated that p(u) can be calculated from the equation ln(p(u))=-alphaT(-1)+beta. The constants alpha and beta are independent of the temperature of the isotherms but they are very sensitive to the structure of adsorbents and to the quality of adsorptives. This fact also makes it possible to characterize exactly the nanostructure of solids and the state of adsorptives. Copyright 2000 Academic Press.
Uniform interpretation of gas/solid adsorption
Abstract A new form of Gibbs equation includes the differential function ψ(Θ) having the form ψ(θ) = ψ( p r ) = ( Θ p r )( dp r dΘ ) where pr is the relative equilibrium pressure and Θ is the coverage. This relationship is the physical and mathematical base for a uniform interpretation (UI) of gas/solid physical adsorption. The function ψ(Θ) is the relative free energy of the surface, therefore, it can be applied to define the thermodynamical correctness of adsorption isotherm equations. Two conditions of correctness have been defined. According to these conditions the change in the free energy of the surface must have finite values at Θ → 0 and at pr → 1. It has been proven that most of the known and widely-used isotherm equations are, from thermodynamic standpoint, incorrect. The first aim of UI is, with help of the differential equation ψ(Θ), to transform the incorrect relationships into correct ones. The second aim is to derive new and correct isotherm equations applicable for practical and theoretical purposes. The third practical applicability of UI is that it provides an exact method to choose the appropriate and correct istherm equation for describing measured isotherms. The most important role of UI is in the calculations of the specific surface area (as) from measured isotherms of Type I and II. The UI modifies our old models (Langmuir, Fowler-Guggenheim, Sips, Jovanovich, Volmer, deBoer-Hill, BET etc.) applicable to calculate the specific surface area. This modification has two important consequences. The first one means that in the isotherm equations a new parameter, χ, appears and it has great influence on the value of as. The second one means that with help of function ψ(pr) one can distinguish the monolayer and multilayer domains in isotherms of Type I and II. Furthermore, the uniform interpretation needs a new model, so-called “cloud model”, for interpretation of multilayer adsorption. This model has important theoretical and practical differences in comparison to the well-known BET theory and also provides new possibility to exact calculations of as. Based on the UI it is possible to calculate the mixed gas adsorption from the pure isotherms measured on heterogeneous surfaces and to qualify industrial adsorbents for practical use. As a consequence of UI it was possible to define the theorem of “referable states” of two dimensional layers (gases) wich corresponds to the “corresponding states” of three-dimensional gases. Shortly summarized: the uniform interpretation proposed in this review modifies our old views of classic models of physical adsorption.
A comparison of some simple adsorption isotherms for describing divalent cation adsorption by ferrihydrite
An adsorption model for divalent cation (M2+) adsorption by ferrihydrite (iron hydrous oxide gel) was developed. The model was used to describe the pH and concentration dependence of M2+ adsorption in 1 M NaNO3. The model was derived by initially considering the statistical thermodynamics of H+−M2+ exchange on a uniform surface. The competitive Langmuir isotherm so derived was characterized by three parameters: a site affinity, a maximum adsorption capacity, and a parameter describing the H+/M2+ exchange stoichiometry. The problem of surface heterogeneity was then formulated in terms of the distribution of sites obeying this local isotherm. The distribution was estimated for ferrihydrite by fitting experimental Ca and Zn adsorption data to four closely related composite isotherms, each with a characteristic distribution of site affinities. The Toth isotherm (four adjustable parameters) and two-site Langmuir isotherm (five adjustable parameters) fitted the experimental data best. The resultant isotherms enabled Ca and Zn adsorption to be estimated within a factor of 1.5 or better over a wide range of pH and concentration. The associated site affinity distributions suggest that the surface of ferrihydrite is composed of a relatively large number of low affinity sites and a small number of high affinity sites.