# Activated Carbons as Effective Adsorbents of Non-Steroidal Anti-Inflammatory Drugs
## 2 Materials and Methods
### Methods
#### 2.2.2 Adsorption Kinetics  
The adsorption rate tests were performed with a flow cell, which allows operation in a closed system (Cary 100 UV-Vis spectrophotometer, Varian Inc., Belrose, Australia). The initial concentration of pharmaceuticals was 0.298 mmol/L, and the mass of GAC was 0.05 g. The adsorption measurements were conducted in a thermostated double-walled vessel at 25 \({}^{\circ}\)C and 35 \({}^{\circ}\)C (Ecoline RE 207 thermostat (Lauda, Lauda-Konigshofen, Germany)). The analyzed solution was stirred with a digitally controlled mechanical stirrer (IKA, Raszyn, Poland) at a speed of 110 rpm during the measurements. The applied adsorption rate testing methodology allowed for obtaining from 70 to 120 experimental points for one system.  
Data on the adsorption rate of pharmaceuticals were analyzed using numerous equations and models of adsorption kinetics. The simplest of them is the first-order kinetic equation (FOE)/pseudo-first-order equation (PFOE), which has the form:  
\[\ln\left(c_{\text{eq}}-c\right)=\ln(c_{\text{eq}}-c_{\text{o}})-k_{1} \tag{3}\]  
or  
\[\ln\left(a_{\text{eq}}-a\right)=\text{Ina}_{\text{eq}}-k_{1}t \tag{4}\]  
where \(c\)--temporary concentration, \(a\)--actual adsorbed amount, the \"o\" and \"eq\" indices correspond to the initial and equilibrium values, and \(k_{1}\)--adsorption rate coefficient [24; 25]. The next uncomplicated relation is the second-order equation (SOE)/pseudo-second-order equation (PSOE). It is expressed in the following form:  
\[a=a_{\text{eq}}[k_{2}t/(1+k_{2}t)] \tag{5}\]  
or its linear forms:  
\[t/a=\left(1/a_{\text{eq}}\right)(1/k_{2}+t) \tag{6}\]  
and  
\[a=a_{\text{eq}}-(1/k_{2})(a/t) \tag{7}\]  
where \(k_{2}=k_{2a}a_{\text{eq}}\) and \(k_{2a}\) are the rate coefficients for pseudo-second-order kinetics [25; 26; 27]. Hereafter, the kinetic data were analyzed using a 1,2-mixed-order kinetic equation (MOE), which is a generalization of the first- and second-order kinetics. This dependence (mathematically) corresponds to a linear combination of first- and second-order units. The above-mentioned relationship might also be expressed as the relative progress of adsorption over time F.  
\[F=a/a_{\text{eq}}=\frac{1-\exp(-k_{1}t)}{1-f_{2}\exp(-k_{1}t)} \tag{8}\]  
or  
\[\ln\left(\frac{1-F}{1-f_{2}F}\right)=-k_{1}t \tag{9}\]  
where \(f_{2}<1\)--the normalized share of the second-order process in the kinetics. One should notice that in some cases, the MOE equation may be reduced to the first-order (\(f_{2}=0\)) and the second-order (\(f_{2}=1\)) kinetic equations [28; 29]. The kinetic data were also analyzed by applying a modified MOE equation that takes into account heterogeneity effects. The fractal-like MOE equation (f-MOE) can be presented as:  
\[F=\frac{1-\exp(-k_{1}t)^{P}}{1-f_{2}\exp(-k_{1}t)^{P}} \tag{10}\]  
where p--fractal coefficient. Under particular conditions, the dependence can be reduced to the MOE (p = 0), f-FOE (\(f_{2}=0\)), and f-SOE (\(f_{2}=1\)) equations [30; 31]. The multi-exponential equation (m-exp) was also used for the specification of kinetic data. This relationship is often used to describe adsorption on heterogeneous solids and may indicate the occurrence of a series of first-order or subsequent processes. This can be written as:  
\[c=\left(c_{0}-c_{\text{eq}}\right)\sum_{i=1}^{n}f_{i}\text{exp}(-k_{i}t)+c_{ \text{eq}} \tag{11}\]  
or  
\[c=c_{0}-c_{0}u_{\text{eq}}\sum_{i=1}^{n}f_{i}[1-\exp(-k_{i}t)] \tag{12}\]  
where \"i\"--the term of the m-exp equation. \(k_{i}\)--the rate coefficient. \(u_{\text{eq}}=1-c_{\text{eq}}/c_{0}\)--the relative loss of adsorbate from the solution [8; 26].