# Freundlich Isotherm: An Adsorption Model Complete Framework
## 1 Introduction  
Long ago the Freundlich isotherm found itself among the isotherms through which one typically checks experimental data obtained. The purpose of this action consists most often of the determination of fitting parameters in order to choose on the \(R^{2}\) approximation basis which model out of the de facto standard set (Langmuir-, Temkin-, and maybe, Dubinin-Radushkevich) of models provides the best fit. Should it be the Freundlich model, one could dwell on the degree of energetic inhomogeneity of the surface (the magnitude of parameter \(\alpha\) in Equation (1)):  
\[\theta=Kx^{\alpha} \tag{1}\]  
where \(x\) and \(\theta\) are in bulk concentrations (for solutions with a constant ionic power. As \(c\), pressure in the gas phase or ionic activity in the solution can be considered) of a surface-active substance being adsorbed and related surface coverage, respectively. The power of \(x\) is often considered as \(\alpha=1/n\), where \(n>0\) is an integer or real number, and the coverage is expressed through adsorption \(q\) and limiting adsorption \(q_{\infty}\) as \(\theta=q/q_{\infty}\). More advanced analysis of fitting results gives rise to a conclusion on which type of the surface (in)homogeneity takes place for a pair \"adsorbate/adsorbent\" considered, that is, evenly inhomogeneous (Temkin adsorption model, or AM), exponentially inhomogeneous ([PERSON]) or homogeneous ([PERSON]) (for a more complete set, one could consider a linear inhomogeneity as well [1], but its representation through the proper equation is much more complicated than those for the other three. The Dubinin-Radushkevich AM is to be mentioned again as well but this is more applicable for adsorption on materials with pronounced porous properties). A huge variety of realizations of the Freundlich AM, amongwhose are the works having studied the adsorption of organic and inorganic substances on substrates of different nature in the framework of chemical, physical, biophysical processes including nano-particles as just examples [2; 3; 4; 5; 6; 7; 8; 9; 10], seems to represent a substantial reason for revisiting the basics of the Freundlich AM. While the monograph by [PERSON] first convincingly elucidated the meaning of Equation (1) for interpretation of experimental data [11], the proper theoretical model was developed by [PERSON] (1934) [12,13], and later reproduced as one of parametrical limits while expanding the general solution for energetically heterogeneous surfaces, expressed through a hypergeometric function into an infinite series [14,15]. The limiting adsorption is bounded through the surface available and an effective number of layers (should it be multi-layer). As it follows from the functional form itself of Equation (1), it does not describe a saturation regime, so that deviations from Equation (1) appear at bigger \(x\). It has become typical for the research community, that data obtained and to be analysed in the framework of Freundlich AM, are to be typically checked for their relevance to Equation (1) with a successive determination of constants \(K\) and \(a\) as fitting parameters. Nevertheless, [PERSON]'s theory of the Freundlich AM goes beyond the small and moderate coverages and mathematically describes the saturation regime too. This circumstance seems to be most typically either ignored or forgotten by the research community although studies with the employment of the Freundlich isotherm for pairs \"adsorbate/adsorbent\" appear to be published in hundreds (if not more) of articles annually. On the basis of [PERSON]'s theory, it appears possible to link the area of small coverages and that at saturation. Furthermore, the theory is applicable to both mono- or multi-layer adsorption. In the last case, one has to distinguish between successive adsorption of layers \"one after one\" like a staircase and simultaneous adsorption when successive layers are being filled without completely filling the foregoing ones. For the former, the theory is to be applied to each \"stair\" separately.