## 2 Results and Discussion
### Equilibrium Studies--adsorption Isotherms  
In this paper, we studied the adsorption of sulfamethoxazole and tetracycline from aqueous solutions using two-parameter isotherms, Langmuir and Freundlich, and three-parameter isotherms, Sips and Redlich-Peterson. For precise calculations of the isotherms characteristic parameters we have used the non-linear form of the isotherms (in order to avoid inherent errors due to linearization) and non-linear fitting of the experimental equilibrium data. The _Langmuir_ isotherm [47], although initially introduced to describe the adsorption of the gases on the solid adsorbents, was subsequently used extensively for the adsorption at the liquid-solid interfaces. The isotherm assumes that the adsorption surface is homogeneous, with active adsorption centers that are identical and uniformly distributed, and with no interaction between the vicinal centers. Each adsorption center can accommodate a single adsorbate molecule. The model considers that the adsorption rate is proportional to the free fraction of the adsorbent and that the desorption rate is proportional to the fraction of the adsorbent occupied. At equilibrium, the two rates become equal. The non-linear form of the _Langmuir_ isotherm is presented in Equation (4):  
\[q_{e}=q_{\max}\frac{K_{L}\cdot C_{e}}{1+K_{L}\cdot C_{e}} \tag{4}\]  
where \(q_{e}\) is the equilibrium adsorption capacity of the mM\(\cdot\)g\({}^{-1}\) adsorbent, \(C_{e}\) is the equilibrium molar concentration of the adsorbate in the solution, mM\(\cdot\)L\({}^{-1}\), \(K_{L}\) is the Langmuir equilibrium constant L\(\cdot\)mM\({}^{-1}\), and \(q_{max}\) is the maximum adsorption capacity of the mM g\({}^{-1}\) adsorbent. The essential features of the Langmuir isotherm can be expressed in terms of the dimensionless constant \(R_{L}\), the separation factor:  
\[R_{L}=\frac{1}{1+K_{L}\cdot C_{0}} \tag{5}\]  
Based on the values of the separation factor, the adsorption is considered _unfavorable_ if \(R_{L}>1\), _linear_ if \(R_{L}=1\), _favorable_ if \(0<R_{L}<1\), and _irreversible_ if \(R_{L}=0\). Figure 10: Plot of Ho–McKay pseudo-second order kinetic equation for the adsorption of sulfamethoxazole and tetracycline, respectively, on PAB3 polymeric adsorbent. The _Freundlich_ isotherm [48] is an empirical adsorption isotherm that is used in adsorption studies on solid adsorbents with heterogenous surfaces and at medium concentrations of adsorbate solutions:  
\[q_{e}=K_{F}\cdot C_{e}^{\frac{1}{6}} \tag{6}\]  
where \(q_{e}\) is the equilibrium adsorption capacity, mM\(\cdot\)g\({}^{-1}\), \(C_{e}\) is the equilibrium molar concentration of the adsorbate in solution, mM\(\cdot\)L\({}^{-1}\), \(K_{F}\) is the Freundlich constant, mM\({}^{(1-1/n)}\cdot\)L\({}^{1/n}\cdot\)g\({}^{-1}\), and \(1/n\) is the constant, depending on the nature and intensity of the adsorption interactions and reflecting the heterogenity degree of the adsorbent surface. The Langmuir and Freundlich isotherms are in fact the two limit adsorption types at the solid-liquid interface. In order to better fit the experimental equilibrium data, three-parameter isotherms were used in order to correct for the inaccuracies eventually induced by the isotherms' linearization. The _Sips_ adsorption isotherm [49], or the combined Langmuir-Freundlich isotherm, is also a three-parameter isotherm assuming cooperative sorbate-sorbate interactions.