# The Role of Clean Coal Technologies in the Development of Renewable Energy Sources
## 1 Introduction
### Membrane Techniques
The plots obtained from the Boyd model were non-linear and none of them passed through the origin, indicating that adsorption is mainly controlled by the diffusion of Pr ions through the liquid film surrounding the adsorbent. In turn, the calculated De value corresponds to values characteristic of chemisorption systems. Based on the analysis of experimental data using the second computer application, it was found that a much better fit was obtained for the Langmuir model (R\({}^{2}\) = 0.848), which indicates a uniform distribution of ion binding sites on the adsorbent surface and exhibits single-layer adsorption features (Table 2). The Dubinin-Radushkevich model is also suitable for describing experimental data (R\({}^{2}\) = 0.958) and is an excellent tool for distinguishing between chemical and physical adsorption of metal ions. Three characteristic parameters of this model, namely \(q\)m, the K\({}_{\rm DR}\) [mol\({}^{2}\)/k\({}^{2}\)] and the average adsorption energy E [k]/mol] were calculated. As we can see, the value of \(q\)m is lower than that determined using the Langmuir model, the value of E, i.e., 122.188 [k]/mol], is characteristic of the chemisorption process (above 80 kJ/mol). Thus, considering the above-mentioned information, it can be concluded that the binding sites are evenly distributed on the surface of the sorbent, the ion adsorption is mono-layer, and the affinity of the sorbent to praseodymium ions is high. The next stage of the research was the analysis of selectivity for praseodymium ions towards competitive ions in the form of Nd, Gd, Dy and Y, and matrix ions (Na, Mg, Ca, Al, Fe and Si), which are usually present in coal fly ash extracts. The selectivity of the tested materials was assessed using the K\({}_{\rm d}\) parameter, i.e., the distribution coefficient and selectivity coefficients k (Table 3). It was found that the obtained K\({}_{\rm d}\) value (1824.529 mL/g) indicates the high efficiency of the tested membrane in recovering praseodymium ions. This may be due to the chelation mechanism (amine groups have the ability to coordinate Pr\({}^{3+}\) ions) and the synergy of the steric effect of imprinting cavities for Pr\({}^{3+}\) ions (the size of the cavities obtained is specific for individual REE ions), showing a strong adsorption interaction with the analyzed ions. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**REE Ion**} & \multirow{2}{*}{**Model**} & **q\({}_{\rm m}\)** & **K\({}_{\rm t}\)** & **K\({}_{\rm z}\)** & **k\({}_{\rm r}\)** & **K\({}_{\rm rp}\)** & **K\({}_{\rm rp}\)** & **D\({}_{\rm r}\)** & **m\({}^{2}\)/min** & **R\({}^{2}\)** \\ \cline{3-10}  & & **[mg/g]** & **[min\({}^{-1}\)]** & **[gg\({}^{-1}\)min\({}^{-1}\)]** & & & **[mg/g]** & **[m\({}^{2}\)/min]** & **R\({}^{2}\)** \\ \hline \multirow{4}{*}{Pr} & I pseudo-order & 13.206 & 0.012 & & & & & & 0.473 \\ \cline{2-10}  & II pseudo-order & 44.248 & & 0.004 & & & & & 0.995 \\ \cline{2-10}  & \multicolumn{2}{c}{Elouch} & & & 6.014 & 5.561 & & & & 0.856 \\ \cline{2-10}  & \multicolumn{2}{c}{Intraparticle} & & & & 1.302 & 14.908 & & & 0.610 \\ \cline{2-10}  & \multicolumn{2}{c}{Diffusion-chemisorption} & 49.751 & & & & 9.842 & & 0.953 \\ \cline{2-10}  & \multicolumn{2}{c}{Boyd} & & & & & & 7.872 \(\times\) 10\({}^{-12}\) & 0.499 \\ \hline \hline \end{tabular}  
* q\({}_{\rm e}\)—adsorption capacity at equilibrium state; K\({}_{\rm t}\) and K\({}_{\rm 2}\)—rate constant of I and II pseudo-order; a and b—Elovich parameters; K\({}_{\rm p}\)—intraparticle diffusion rate; x\({}_{\rm i}\)—intercept; K\({}_{\rm DC}\)—diffusion chemisorption rate; and D\({}_{\rm e}\)—diffusion coefficient.  
\end{table} 
Table 1: Comparison of characteristic kinetic parameters from the pseudo-first and pseudo-second order [PERSON], [PERSON] kinetic model and intraparticle diffusion model, diffusion-chemisorption, and the Boyd model.