# Freundlich Isotherm: An Adsorption Model Complete Framework
## 3 The Consistent Algorithm of Analysis of Experimental Data  
According to the above said, dependencies like Equation (1) appear to be determined and carefully studied on the basis of small and/or moderate concentrations, which include determination of constants \(K\) and \(\alpha\) (blue curves in Figure 1). The saturation regimes (4) (shown grey in Figure 1) turn out to be, as a rule, omitted both in experimental studies and discussions of obtained results. Nevertheless, there are a lot of evident reasons why such regimes, as well as intermediate regions (orange in Figure 1), can represent considerable interest for the research community and do. Therefore, in _step 1_, upon the determination of the fitting parameters \(K\) and \(\alpha\), one is expected to determine the saturation regime. It has the Langmuir-like form \(f_{3}(x)=\frac{x}{x+b}\) where \(b\equiv b_{0}/\left(1+\frac{1}{\alpha}\right)\) (we now redefine constant \(b\) as different from that in Equation (2)). Thus, the use of Equations (4) and (5) enables one to perform this without an introduction of a rigorous boundary between the saturation regime (grey curves) and the intermediate coverage area (orange). Such a boundary (a point between solid and dashed grey curves in Figure 1) will be imposed later. _Step 2_ is related to the determination of the intermediate area with its boundaries. Technically, the most obvious way to model the intermediate region of concentrations would be to perform some interpolation. One would need to identify boundaries \(x_{m}\) and \(x_{s}\) of the intermediate area and to perform linkage on the class of continuously differentiable functions. Whereas boundary \(x_{m}\) is naturally given as the experimental data point (\(x_{m}\), \(f_{1m}\)) for the biggest concentration (it depends on an experimentalist's judgement whether the power law area has not yet been left there), point \(x_{s}\) is to be determined on the basis of some consideration, at which \(x\) the saturation regime begins. This would be a relative judgement. To avoid this, one can properly determine \(x_{s}\) within the method itself linking the intermediate and saturation regime areas. We employ in the intermediate area a quadratic form \(f_{2}(x)\) to be continuously linked with the adsorption function \(f_{1}(x)\) in the area of small and moderate coverages on the left and with adsorption function \(f_{3}(x)\) in the saturation regime area on the right. Out of parabolic, hyperbolic, and elliptic forms we chose parabola whose upper branch in a Cartesian coordinate system with apostrophe hasthe canonical form \({f_{2}^{\prime}}^{2}=2 px^{\prime}\). Accompanying the linking point arbitrary for a while with sub/superscript \"0\", we use the expression for a tangent line to our parabola at this point:  
\[{f_{2}^{\prime}}{f_{2}^{(0)}}^{\prime}=p\big{(}x^{\prime}+x_{0}^{\prime}\big{)} \tag{6}\]  
where \({f_{2}^{(0)}}^{\prime}\equiv{f_{2}^{\prime}}(x_{0}^{\prime})\). In order to continuously differentiably adjust the parabola to \({f_{1}}(x)\) and \({f_{3}}(x)\) at \(x_{m}\) and \(x_{s}\), respectively, we apply a translational transformation \(x^{\prime}=x+c\), \(f_{2}^{\prime}=f_{2}+d\), where \(c\) and \(d\) are constants to be determined later, to Equation (6):  
\[f_{2}(x)=\frac{p}{f_{2}^{(0)}+d}x+p\frac{x_{0}+2c}{f_{2}^{(0)}+d}-d \tag{7}\]  
where \({f_{2}^{(0)}}\equiv{f_{2}}(x_{0})\). At \(x_{0}=x_{m}\), the tangent line (7) is expected to coincide with \({f_{1}}(x)\)'s tangent line.