## 3 Experimental and Calculation Procedures
### Investigation Methods
#### 3.3.6 Adsorption Studies  
Kinetic measurements were conducted using a UV-Vis spectrophotometer Cary 100 (Varian, Australia) with a flow working cell for periodic measurements of a solution concentration in a closed system. In all cases, 100 mg of sample was placed in a quartz vessel connected with a stirrer (110 rpm) and thermostatic system (25 \({}^{\circ}\)C) and filled with 100 mL solutes. Each of the samples was treated with three aqueous solutions of organic compounds: phenol, 4-nitropenol, and nitrobenzene. The initial concentration of solutions in all cases was equal to 0.205 mmol/L. At definite time intervals, the solution samples were gathered in a flow cell, and absorption spectra were measured, and the solution was turned back to the reaction vessel. The absorbance spectrum was recorded for wavelength ranges from 200 to 450 nm. The concentration in the function of time profiles for the experimental systems was calculated from the recorded spectra. The kinetic curves were analysed using several equations: first-order equation (FOE), the second-order equation (SOE) and pseudo first and second-order equations (PFOE and PSOE), the so-called mixed 1,2-order equation (MOE), the fractal-like MOE equation (f-MOE), the fractal-like first-order equation (f-FOE) and the fractal-like second-order equation (f-SOE), the so-called multiexponential equation (m-exp), the intraparticle diffusion model (IDM), and the pore diffusion model (PDM). The basic and pseudo first-order equations (FOE and PFOE can be expressed by a linear relationships (3) or (4):  
\[ln\big{(}a_{eq}-a\big{)}=ln(a_{eq}-a_{o})-k_{1} \tag{3}\]  
where a is the temporary adsorbed amount, \(a_{o}\) is the initial amount, \(a_{eq}\) is the equilibrium adsorbed amount, \(k_{1}\) is the adsorption rate coefficient. \[ln\big{(}c_{eq}-c\big{)}=ln(c_{eq}-c_{o})-k_{1} \tag{4}\]  
where \(c\) is the temporary adsorbate concentration, \(c_{o}\) is the initial concentration, \(c_{eq}\) is the equilibrium concentration, \(k_{1}\) is the same as in Equation (3). The second-order and pseudo second order equations (SOE and PSOE) can be noted as:  
\[a=a_{eq}[k_{2}t/(1+k_{2}t)] \tag{5}\]  
The linear forms of SOE and PSOE equations are:  
\[t/a=\big{(}1/a_{eq}\big{)}(1/k_{2}+t) \tag{6}\]  
\[a=a_{eq}-(1/k_{2})(a/t) \tag{7}\]  
where \(k_{2}=k_{2a}a_{eq}\) and \(k_{2a}\) are the rate coefficients for pseudo-second order kinetics. The generalization of the first and second order kinetics is the 1,2-mixed-order kinetic equation (MOE) which is the combination of the first and second order units and can be expressed as a relative adsorption progress in time (\(F\)):  
\[F=a/a_{eq}=\frac{1-exp(-k_{1}t)}{1-f_{2}exp(-k_{1}t)} \tag{8}\]  
\[ln\bigg{(}\frac{1-F}{1-f_{2}F}\bigg{)}=-k_{1}t \tag{9}\]  
where \(f_{2}<1\) is the normalized share of the second order process in the kinetics. In some cases, the MOE equation is reduced to the simple kinetic equations of the first (\(f_{2}=0\)) and the second order (\(f_{2}=1\)) type. Non-ideality effects can be considered using fractal-like equations (f-FOE, f-SOE and f-MOE) in which fractal coefficient f is used. The multi-exponential equation (m-exp) is applied to describe numerous first-order processes or the follow up processes and has the following forms:  
\[c=\big{(}c_{0}-c_{eq}\big{)}\sum\
olimits_{i=1}^{m}f_{i}exp(-k_{i}t)+c_{eq} \tag{10}\]  
\[c=c_{0}-c_{0}u_{eq}\sum\
olimits_{i=1}^{m}f_{i}[1-exp(-k_{i}t)] \tag{11}\]  
where \(k_{i}\) is the rate coefficient, \(u_{eq}=1-c_{eq}/c_{o}\) is the relative loss of adsorbate from the solution, and \"\(i\)\" is the term of m-exp equation. The Intraparticle Diffusion Model (IDM, Crank) characterizes the adsorption processes on the spherical adsorbent grains.