## 3 Results and Discussion
### Solid-Phase Extraction of Dyes
Considering the change in concentrations in the real-time interval and depending on the dose of the sorbent, Equation (6) can be written as  
\[\mathrm{K}=\frac{(C_{0}-C_{e})V}{mC_{e}} \tag{7}\]  
The Gibbs equation can also be expressed as follows:  
\[\Delta G^{0}=\Delta H^{0}-\mathrm{T}\Delta S^{0} \tag{8}\]  
Combining the above equations, we obtain  
\[lnK=\frac{\Delta S^{0}}{R}-\frac{\Delta H^{0}}{RT} \tag{9}\]  
The obtained thermodynamic parameters are presented in Figure 6:0  
The data presented in Table 2 indicate that the adsorption process occurs spontaneously. The process is exothermic because the enthalpy change is negative. A positive entropy change value indicates that the number of degrees of freedom at the solid-liquid interface increases during dye adsorption and reflects the sorbate affinity for sorbent molecules. The values of the Gibbs free energy change also increase. This suggests that an increase in temperature indicates a higher adsorption efficiency at moderately low temperatures. To achieve equilibrium, it is necessary to provide an appropriate contact time between the dye and the sorbent, as this is one of the main factors in the SPE process based on equilibrium. The results are shown in Figure 6:1 It can be seen that the degree of adsorption is higher at a lower temperature. However, all three temperatures can be considered satisfactory since the degree of adsorption for each of them exceeds 90%. The dye recovery rates during adsorption turned out to be quantitative: SY \(\geq\) 80%; TAR \(\geq\) 90%; Orange II \(\geq\) 93%; Methyl Orange \(\geq\) 85%. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Dye** & **T, K** & **lnK** & **\(\Delta\)G, kJ/mol** & **\(\Delta\)H, kJ/mol** & **\(\Delta\)S, kJ/mol K** \\ \hline \multirow{4}{*}{SY} & 278 & 2.55 & \(-\)5.89 & & \\ \cline{2-6}  & 293 & 2.28 & \(-\)5.56 & \(-\)0.700 & 2.02 \\ \cline{2-6}  & 308 & 2.11 & \(-\)5.40 & & \\ \hline \multirow{4}{*}{TAR} & 278 & 3.21 & \(-\)7.41 & & \\ \cline{2-6}  & 293 & 2.95 & \(-\)7.18 & \(-\)0.753 & 2.80 \\ \cline{2-6}  & 308 & 2.89 & \(-\)7.40 & & \\ \hline \multirow{4}{*}{Orange II} & 278 & 4.01 & \(-\)9.26 & & \\ \cline{2-6}  & 293 & 3.75 & \(-\)9.13 & \(-\)0.674 & 3.30 \\ \cline{2-6}  & 308 & 3.36 & \(-\)8.60 & & \\ \hline \multirow{2}{*}{Methyl Orange} & 273 & 2.55 & \(-\)5.89 & & \\ \cline{2-6}  & 293 & 2.47 & \(-\)6.02 & \(-\)0.809 & 2.405 \\ \cline{2-6}  & 308 & 2.43 & \(-\)6.21 & & \\ \hline \end{tabular}
\end{table} 
Table 2: Calculated thermodynamic parameters. The mechanism of adsorption, particularly the chemical reaction and mass transfer, was analyzed using the pseudo-first-order rate equation and the pseudo-second-order rate equation. The linear form of the pseudo-first-order kinetic model can be expressed using the following equation:  
\[\log(q_{e}-q_{t})=\log q_{e}-(\frac{k_{1}}{2303})t \tag{10}\]  
where \(q_{e}\) (mg/g) is the amount of dye adsorbed at equilibrium; \(q_{t}\) (mg/g) is the amount of dye adsorbed at any given time (\(t\)); and \(k_{1}\) (1/min) is the pseudo-first-order absorption rate constant, which can be calculated from the plot of the logarithm (\(q_{e}-q_{t}\)) versus time. The pseudo-second-order kinetic model can be expressed in linear form using Equation (11):  
\[\frac{t}{q_{t}}=\frac{1}{k_{2}q_{e}^{2}}+\frac{t}{q_{e}} \tag{11}\]  
where \(k_{2}\) (g/(mg/min)) is the pseudo-second-order sorption rate constant, which can be calculated from a linear plot of \(t/q_{t}\) versus time. The kinetic parameters under different conditions are given in Table 3. The applied pseudo-first-order equation satisfactorily describes the patterns of dye adsorption at the initial stages of the adsorption process when the film diffusion phenomenon has a significant effect on the process.