# Metachrylate-Based Polymeric Sorbents for Recovery of Metals from Aqueous Solutions
## 5 Adsorption of Metal Ions on Polymeric Sorbents--General Remarks
### Adsorption Isotherms Models  
By definition, adsorption isotherm is the relationship between the adsorbate in the liquid phase and the adsorbate adsorbed on the adsorbent surface at equilibrium at a constant temperature. Modeling experimental data from adsorption processes is a valuable method for determining potential interactions between adsorbents and adsorbates and predicting the mechanisms in diverse adsorption systems [70; 71]. Adsorption isotherms are mathematical models that illustrate the distribution of metals between adsorbate and adsorbent. Equilibrium data obtained from initial concentration could be analyzed using numerous mathematical isotherm models, reported in the literature in detail [71]. For a single-component system, the most commonly used are two parameters, Langmuir, Freundlich, and Temkin isotherm models, or three-parameters Redlich-Peterson, Sips, and Toth isotherms [70]. The most frequently used adsorption isotherm models are listed in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & **Non-Linear Form** & **Linear Form** & **Parameters** \\ \hline Langmuir & \(q_{t}=\frac{q_{max}K_{t}C_{e}}{1+K_{t}C_{e}}\) & \(\frac{C_{t}}{q_{e}}=\frac{1}{K_{t}q_{max}}+\frac{C_{e}}{q_{max}}\) & \(q_{max}\)—maximum monolayer coverage capacity, \\  & & \(K_{t}\)—Langmuir isotherm constant (binding energy of adsorption) \\ \hline [PERSON] & \(q_{e}=K_{F}C_{e}^{1/n}\) & \(\log q_{e}=\log K_{F}+\frac{1}{n}\log C_{e}\) & \(n\)—Freundlich constant related to adsorption intensity \\ \hline Temkin & \(q_{e}=\frac{RT}{b_{T}}\ln(A_{T}C_{e})\) & \(q_{e}=\frac{RT}{b_{T}}\ln A_{T}+\frac{RT}{b_{T}}\ln C_{e}\) & \(A_{T}\)—Temkin isotherm equilibrium binding constant \\ \hline Elovich & \(\frac{q_{e}}{q_{m}}=K_{E}C_{e}e^{\frac{\alpha_{e}}{\alpha_{e}}}\) & \(\ln\frac{q_{e}}{C_{e}}=\ln K_{E}q_{m}-\frac{q_{e}}{q_{m}}\) & \(K_{E}\)—Elovich equilibrium constant, \(q_{m}\)—Elovich maximum adsorption capacity \\ \hline Dubinin–Radushevich & \(q_{e}=q_{DR}e^{-A_{DR}\epsilon_{2}}\) & \(\ln q_{e}=\ln q_{DR}-A_{DR}\epsilon^{2}\) & \(q_{DR}\)—Dubinin–Radushevich maximum adsorption capacity, \(A_{DR}\)—constant representing mean adsorption energy \\ \hline \hline \end{tabular}
\end{table}
Table 1: Two and three parameters adsorption isotherm models (Nonlinear and linear form) [70; 72; 73; 74; 75].  
Real wastewaters are multi-component systems that contain various pollutants, making the adsorption system more complicated. Thus, interaction and competition between the adsorbate molecules must be taken into account. In order to understand the adsorption mechanism of such complex systems, single-component models were modified. Consequently, non-modified, modified, and extended Langmuir and Freundlich model, Redlich-Peterson model, Sheindorf-Rebuhn-Sheintuch equation, and extended Sips isotherm model were developed [76].
The linear regression correlation coefficient (\(R^{2}\)) values are frequently compared to evaluate the best fit model. However, to assess the best kinetic fitting model, besides regression coefficient (\(R^{2}\)), statistical error validity models such as average relative error, normalized standard deviation, hybrid fractional error function, a derivative of Marquardt's percent standard deviation, and standard deviation of relative error should also be used [71]. Another criterion that should be taken into account is the closeness, i.e., the agreement between the experimental (\(Q_{e}^{exp}\)) and calculated (\(Q_{e}^{calc}\)) value of adsorption capacity.