## 2 Materials and Methods
### Methods and Calculations
#### 2.4.3 Other Techniques and Calculations
\begin{table}
\begin{tabular}{c c c} \hline
**Kinetic Equation** & **Short Description** & **Kinetic Equations** \\
**(Abbreviation)** & & \(\pi=a_{\text{eq}}[k_{2}t/(1+k_{2}t)]\) (\(\beta\)) \\ Second-order equation and & Suitable when concentration changes proceed & or \(t/a=(1/a_{\text{eq}})(1/k_{2}+t)\) and \\ pseudo-second-order equation (SOE and PSOE) & \((a_{\text{eq}}-a)^{2}\)[43; 44]. & \(a_{\text{eq}}[k_{2}t/(1+k_{2}t)]\) \\  & & \(k_{2}=k_{2}+a_{\text{eq}}\) and \(k_{2}\) are the rate coefficients \\  & & for pseudo-second-order kinetics [43]. \\  & & Pseudo-second-order equations may be used for \\  & & \(F=a/a_{\text{eq}}=\frac{1-\alpha\varphi(-k_{1}t)}{1-\lambda\cdot\rho\cdot(-k_{1}t )}\) or \(ln\left(\frac{1-F}{\lambda^{-2}t^{2}}\right)=-k_{1}t\) (4) \\  & & where \(F\) is the relative adsorption progress in time, \(f_{2}<1\) is the normalized share of the second-order process in the kinetics. In special cases, the MOE equation may be degraded to the simple kinetic equations of the first (\(f_{2}=0\)) and the second order (\(f_{2}=1\)) type [46; 47]. \\ \hline Fractal-like MOE equation (f-MOE) & \begin{tabular}{c} Includes probably distribution of rate \\ coefficients. Adequate description of the \\ non-ideality effects [48]. \\ \end{tabular} & \(F=\frac{1-\alpha\varphi(-k_{1}t)^{\alpha}}{1-\lambda\cdot\rho\cdot(-k_{1}t)^{ \alpha}}\) (5) \\  & & where \(p\) is the fractal coefficient. \\ \hline Multiexponential equation & \
\begin{tabular}{c} Generalization of Lagergren equation to a \\ series of parallel first-order processes [49; 50]. \\ Right for a description of the adsorption \\ kinetics on energetically heterogeneous solids \\ that could not be expressed by the \\ FOE/SOE/MOE equations [51]. \\ \end{tabular} & \(c=\left(c_{\text{eq}}-c_{\text{eq}}\right)\sum_{i=1}^{n}f_{\text{f}}exp(-k_{ i}t)+c_{\text{eq}}\) (6) \\  & & or \(c=c_{\text{eq}}-c_{\text{eq}}\omega_{\text{eq}}\sum_{i=1}^{n}f_{i}[1-exp(-k_{ i}t)]\) \\  & & where \(``\
u\"\) is the term of m-exp equation, \(k_{i}\) is the \\  & & rate coefficient and \(\text{u}_{\text{eq}}=1-c_{\text{eq}}/c_{\text{eq}}\) is the relative \\  & & loss of adsorbate from the solution. \\ \hline Intraparticle Diffusion Model & Adsorption on the spherical adsorbent & \(F=1-\frac{\beta}{\pi^{2}}\sum_{i=1}^{n}\frac{\beta}{\pi^{2}}exp\left(-\frac{ \pi^{2}\cdot p_{\text{eq}}\cdot D_{\text{eq}}t}{\pi^{2}}\right)\) (7) \\  & & where \(r\) is the radius of the adsorbent particle, \(D_{\text{eq}}\) is the effective diffusion coefficient: \\ (IDM, Crank) & & \(D_{\text{eq}}=\frac{D}{\gamma_{p}\cdot\left(1+p_{\text{eq}}K_{\text{eq}} \right)p}\) (8) \\  & & where \(D\) is the molecular diffusion coefficient, \(\tau_{p}\)-the dimensionless pore tortuosity factor, \(\rho\) is the particle density, \(\epsilon_{p}\) is the particle porosity, \(K_{\text{ff}}\)-the \\  & & [PERSON] adsorption constant. \\ \hline \multirow{7}{*}{
\begin{tabular}{c} Pore Diffusion Model \\ (PDM, McKay) \\ \end{tabular} } & \(\frac{dF}{2\tau_{\text{eq}}}=\frac{3\left(1-u_{\text{eq}}F\right)\cdot\left(1-F \right)^{\frac{1}{2}}}{1-B\cdot\left(1-F\right)^{\frac{1}{2}}}\) (9) \\  & & where \(u_{\text{eq}}\) is the relative adsorbate loss, the \\ \cline{1-1}  & & \(B=1-1/B_{\text{eq}}\), where \(B_{\text{eq}}=K_{\text{ff}}D_{\text{eq}}\) is the \\ \cline{1-1}  & & \(B=\left(\frac{1}{1-u_{\text{eq}}}\right)^{\frac{1}{2}}\) (10) \\ \cline{1-1}  & & where \(x=\left(1-F\right)^{\frac{1}{2}}\), \(b=\left(\frac{1}{1-u_{\text{eq}}}\right)^{\frac{1}{2}}\) (11) \\ \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)Germany) atomic absorption spectrometer was applied for silver content determination in the initial colloidal silver nanoparticles solution.