# Metachrylate-Based Polymeric Sorbents for Recovery of Metals from Aqueous Solutions
## 5 Adsorption of Metal Ions on Polymeric Sorbents--General Remarks
### Adsorption Thermodynamics  
One of the important parameters of the adsorption process is temperature, i.e., the endothermal and exothermic character of the process, determined by the increase or decrease of the temperature all through the adsorption process. In order to evaluate the feasibility of the adsorption process, thermodynamic parameters such as the standard free energy (\(\Delta G^{0}\)), enthalpy change (\(\Delta H^{0}\)), and entropy change (\(\Delta S^{0}\)) were estimated. The Gibb's free energy change of adsorption was calculated from the following equation:  
\[\Delta G^{0}=-RT\ln K_{c} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & **Non-Linear Form** & **Linear Form** & **Parameters** \\ \hline Pseudo-first-order & \(q_{t}=q_{e}\left(1-e^{-k_{1}t}\right)\) & \(\log(q_{e}-q_{t})=\log(q_{e})-\frac{k_{1}}{2.303}t\) & \(q_{e}\)—adsorption capacity at equilibrium, \(k_{1}\)—pseudo-first-order rate constant \\ \hline Pseudo-second-order & \(q_{t}=\frac{q_{e}^{2}k_{2}t}{1+q_{t}k_{2}t}\) & \(\frac{t}{t!}=\frac{1}{k_{2}q_{t}^{2}}+\frac{t}{t! }\) & \(k_{2}\)—pseudo-second-order rate constant \\ \hline Elovich & \(q_{t}=\frac{1}{p}\ln(\alpha\beta t+1)\) & \(q_{t}=\frac{1}{p}\ln(\alpha\beta)+\frac{1}{p}\ln(t)\) & \(\alpha\)—initial adsorption rate, \(\beta\)—desorption constant \\ \hline Avrami & \(q_{t}=q_{e}(1-\exp[-k_{AV}t^{\mu}])\) & \( \begin{matrix}\ln\left(\ln\left(\frac{q_{e}}{q_{e}-q_{t}}\right)\right)\\ =n\ln k_{AV}+n\ln t\end{matrix} \) & \( \begin{matrix}k_{AV}\text{--Avrami fractional-order}\\ \text{constant rat, }n\text{--Avrami fractional}\\ \text{kinetic order}\end{matrix} \) \\ \hline Fractional power & \(q_{t}=Kt^{v}\) & \(\log q_{t}=\log K+v\log t\) & \(K\)—fractional power constant, \(v\)—fractional power rate constant \\ \hline Intraparticle diffusion & \(q_{t}=k_{id}t^{0.5}+C_{id}\) & \(k_{id}\)—intraparticle diffusion rate constant, \(C_{id}\)—thickness of the adsorbent \\ \hline Liquid film diffusion & \(q_{t}=q_{e}(1-e^{-k_{LTD}t})\) & \( \begin{matrix}\ln(1-F)=-k_{LTD}t+C,F\\ =\frac{q_{t}}{q_{e}}\end{matrix} \) & \( \begin{matrix}F\text{--fraction of solute}\\ \text{adsorbed at any time }t,k_{LTD}\\ \text{equilibrium fractional attainment}\end{matrix} \) \\ \hline Bangham & \(q_{t}=k_{B}t^{\mu}\) & \( \begin{matrix}\log\log\left(\frac{C_{0}}{C_{0}-q_{t}\mu}\right)\\ =\log\frac{k_{B}m}{2.3037}+\alpha\log t\end{matrix} \) & \(k_{B}\)—constant rate of Bangham model, \(\alpha\)—constant (indicates the adsorption intensity) \\ \hline Boyd & \( \begin{matrix}F\textless 0.85\\ q_{t}=q_{e}\frac{3}{\pi}\left[1-\left(1-\left(\frac{(Bt)^{2}}{\sqrt{\pi}} \right)^{2}\right)\right]\\ F>0.85\\ q_{t}=q_{e}\left(1-\frac{6}{\pi^{2}}e^{(-Bt)}\right)\end{matrix} \) & \( \begin{matrix}F\textless 0.85\\ Bt=\left(\sqrt{\pi}-\sqrt{1-\frac{\pi F}{3}}\right)^{2}\\ F>0.85\\ Bt=-0.4977-\ln(1-F)\end{matrix} \) & \(\begin{matrix}F\textless 0.85\\ Bt=-\left(\sqrt{\pi}-\sqrt{1-\frac{\pi F}{3}}\right)^{2}\\ Bt=-\left(\sqrt{\pi}\right)\text{where \(R\) is the ideal gas constant (8.314 J/mol K), \(T\) (K) is the absolute temperature and \(K_{c}\) is the thermodynamic equilibrium constant that is expressed as:  
\[K_{c}=\frac{C_{a}}{C_{e}} \tag{3}\]  
where \(C_{a}\) (mg/L) is the amount of metal ion adsorbed at equilibrium, and \(C_{e}\) (mg/L) is the concentration of metal ions in solution at equilibrium. [PERSON]'s free energy is also related to the enthalpy change and entropy change at constant temperature by the Van't Hoff equation as follows [83; 84]:  
\[\ln K_{c}=-\frac{\Delta G^{0}}{RT}=\frac{\Delta S^{0}}{R}-\frac{\Delta H^{0}}{ RT} \tag{4}\]  
The \(\Delta H^{0}\) and \(\Delta S^{0}\) values can be calculated from the slope and intercept of the plot of ln \(K_{c}\) versus \(1/T\). The negative values of \(\Delta G^{0}\) indicate that the adsorption process is spontaneous. In the case of positive entropy change, the randomness at the moment of adsorption rises, while in the case of negative free energy change, adsorption is spontaneous or favorable.