# Ibuprofen Removal by Aluminum-Modified Activated Carbon (AC@Al) Derived from Coconut Shells
## 2 Materials and Methods
### Adsorption Experiments
#### 2.4.1 Equilibrium Experiments  
For isothermal equilibrium tests, a fixed amount of the AC@Al adsorbent (g) was added to 15 mL falcon tubes containing 10 mL of the IBU solution (30 to 200 mg/L). For the evaluation of the results, Langmuir and Freundlich isotherm models were applied. Equation (3) is related to the Langmuir model [20]:  
\[Q_{e}=\frac{Q_{m}K_{L}C_{e}}{1+K_{L}C_{e}} \tag{3}\]  
where \(Q_{e}\) is the concentration of IBU adsorbed in the solid phase (mg/g), \(Q_{m}\) is the maximum adsorption capacity (mg/g), and \(K_{L}\) is the relative energy for IBU adsorption(L/mg).  
Initially, the [PERSON] and Freundlich models were applied in this study. In the Langmuir model, the energy of adsorption is constant, as is the number of available adsorption sites on the surface of the adsorbent. Each site can only be occupied by one molecule, resulting in the formation of a monolayer with no interactions between the adsorbed molecules. Essentially, the adsorbent has a finite adsorption capacity (\(Q_{m}\)), that is, the maximum amount that the AC@Al surface can adsorb in equilibrium. The Freundlich model [21] deals with both monolayer and multilayer adsorption on heterogeneous surfaces. It describes the relationship regarding the concentration of IBU at equilibrium (mg/L) and the relative adsorption capacity of AC@Al, \(Q_{e}\) (mg/g), as stated in Equation (4).  
\[Q_{e}=K_{F}C_{e}^{1/n} \tag{4}\]  
where \(K_{F}\) is the adsorption's capacity constant and \(1/n\) is the adsorption ability's or surface's heterogeneity constant.  
In addition, as will be demonstrated by the experimental results, it was also necessary to apply the Langmuir-Freundlich (L-F) equation model (also known as the Sips isotherm model [22]) to the data of this study. The Sips isotherm, that is a combination of the Langmuir and Freundlich models, is a model that studies three parameters to adjust experimental results. It is represented by Equation (5):  
\[Q_{e}=\frac{Q_{m}K_{LF}C_{e}^{1/n}}{1+K_{LF}C_{e}^{1/n}} \tag{5}\]  
where \(K_{LF}\) is the equilibrium constant of heterogeneous surfaces (L/g), \(Q_{e}\) is the adsorbed amount at equilibrium (mg/g), \(Q_{m}\) is the maximum adsorption capacity (mg/g), \(C_{e}\) is the adsorbate equilibrium concentration (mg/L) and n is the heterogeneity parameter (0 < \(n\) < 1).  
Equation (5) is used for both high and low solution concentrations. In particular, at low concentrations, this isotherm follows the Freundlich isotherm, and at high concentrations, it follows the Langmuir isotherm, predicting monolayer adsorption capacity [23].  
Finally, according to several researchers [24; 25; 26], the accuracy of the Redlich-Peterson (R-P) isotherm equation, including three adjustable parameters, is better than the Langmuir and Freundlich counterparts. Thus, the Redlich-Peterson model is widely used as the link between Langmuir and Freundlich isotherm models. The equation applied for this model is Equation (6):  
\[Q_{e}=\frac{K_{RP}C_{e}}{1+\alpha_{R}C_{e}^{n}} \tag{6}\]  
where \(K_{RP}\) (L/g) and \(\alpha_{RP}\) (L/mg)\({}^{n}\cdot 10^{3}\)) are the R-P constants and \(n\) reflects adsorbent heterogeneity.