# Application of Three-Dimensional Porous Aerogel as Adsorbent for Removal of Textile Dyes from Water
## 2 Materials and Methods
### Adsorption Isotherms and Kinetics  
Isothermal and kinetic studies play an important role in estimating the behavior and performance of an adsorption system [21]. Adsorption isotherms describe the phenomenon that regulates the movement of molecules from the mobile phase to the solid phase at a constant temperature [22].  
Most of the works on aerogels have suggested that the Langmuir isotherm is better suited for representing the data in comparison to other models, including the Temkin or Freundlich model [23].  
The adsorption process equilibrium data were fitted to the Langmuir and Freundlich isotherms. The equilibrium relation between the adsorbed amount of the adsorbate and the amount of the adsorbate in the solution was determined by using the following equation:  
\[q_{e}=\frac{\left(\begin{array}{cc}C_{0}-C_{e}\end{array}\right)V}{m}, \tag{1}\]  
where \(\mathrm{q_{e}}\) (mg/g)--adsorption capacity; \(\mathrm{C_{0}}\) (mg/L)--initial concentration of adsorbate; \(\mathrm{C_{e}}\) (mg/L)--equilibrium concentration of adsorbate; V (L)--volume of solution; and m (g)--mass of the adsorbent.  
Langmuir and Freundlich isotherms were chosen as best describing the performance of the adsorbent made from paper and cardboard waste.  
The formula for the Langmuir isotherm is  
\[q_{e}=\frac{q_{m}\,K_{L}C_{e}}{1+K_{L}C_{e}}, \tag{2}\]  
and that for the Freundlich isotherm is  
\[q_{e}=K_{f}C_{e}^{\frac{1}{n}}, \tag{3}\]  
where \(\mathrm{q_{m}}\) is the maximum adsorption capacity (mg/g), \(\mathrm{K_{L}}\) is the [PERSON] isotherm constant (l/g), \(\mathrm{K_{f}}\) is the Freundlich isotherm constant (mg/g)/(mg/l)\({}^{-1/n}\) and \(n\) is the Freundlich exponent.  
In equilibrium studies, it is customary to use linear forms of isotherms.  
The linear form of the Langmuir isotherm used is  
\[\frac{C_{e}}{q_{e}}=\frac{1}{q_{m}K_{L}}+\frac{C_{e}}{q_{m}}. \tag{4}\]  
And the linear form of Freundlich isotherm is  
\[\log q_{e}=\log K_{f}+\frac{1}{n}\log C_{e}. \tag{5}\]  
If we use the Freundlich isotherm, then the Freundlich exponent (\(n\)) is the adsorption process characteristic. Exponent values above 1 have a favorable adsorption process and values less than 1, an unfavorable process.  
In order to understand the mechanism of the adsorption process, the experimental data were fitted to the following well-known kinetic models: the pseudo-first-order and the pseudo-second-order models. The pseudo-first-order model equation is  
\[q_{t}=q_{e}(1-exp(-k_{1}t)), \tag{6}\]  
and the linear form of the pseudo-first-order model is  
\[ln|q_{e}-q_{t}|=ln|q_{e}|-k_{1}t. \tag{7}\]  
The pseudo-second-order model equation is  
\[q_{t}=\frac{q_{e}^{2}k_{2}t}{1+k_{2}q_{e}t}, \tag{8}\]  
and the linear form of the pseudo-second-order model is  
\[\frac{t}{q_{t}}=\frac{1}{k_{2}q_{e}^{2}}+\frac{t}{q_{e}}, \tag{9}\]where \(q_{e}\) represents the amount of adsorption at equilibrium, \(q_{t}\) represents the amount of adsorption at time t, \(k_{1}\) is the rate constant of the pseudo-first-order model and \(k_{2}\) is the rate constant of the pseudo-second-order model.  
The constants of the pseudo-models were determined in two ways, i.e., by applying linear model forms and by applying nonlinear regression and minimizing the sum:  
\[\sum_{i=1}^{N}\left(q_{t,exp}-q_{t,calc}\right)^{2}, \tag{10}\]  
where \(q_{t,exp}\) is the obtained value in adsorption experiments at time moment t and \(q_{t,calc}\) is the value predicted by using the equation of the studied isotherm at time t.  
The coefficient of determination and the average relative errors were calculated as follows:  
\[R^{2}=1-\frac{\sum_{i=1}^{N}\left(q_{t,exp}-q_{t,calc}\right)^{2}}{\sum_{i=1}^ {N}\left(q_{t,exp}-\overline{q}_{t,exp}\right)^{2}}, \tag{11}\]  
and  
\[ARE=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{q_{t,exp}-q_{t,calc}}{q_{t,exp}} \right|100\%, \tag{12}\]  
where \(\overline{q}_{t,exp}\) is the average of \(q_{t,exp}\), i.e., \(\overline{q}_{t,exp}=\frac{1}{N}\sum\limits_{i=1}^{N}q_{t,exp}^{i}\), and \(N\) is the number of time moments.