## 2 Experimental
### Batch Kinetic Adsorption Isotherm Tests (Single Element)  
A series of single-element batch kinetic rate law experiments were conducted using individual stock solutions of Nd(III), Y(III), Ce(III), Er(III), and Fe(III) in 3 M HCl with concentrations of 93 mg/L, 137 mg/L, 126 mg/L, and 117 mg/L, and 477 mg/mL, respectively. The variation in feed concentration for each element was strategically implemented to ensure that the recovery remains at approximately 50%. This approach guarantees that the resin has an adequate quantity of metal ions available for adsorption. Furthermore, in practical applications, leachates derived from different sources contain a wide range of concentrations for various REEs.  
For the adsorption test, 0.1 g of TK221 was soaked in 5 mL of 3 M HCl in a 50 mL Erlenmeyer flask for at least 12 h at room temperature to allow the resin to swell and increase its capacity. After swelling, 5 mL of the REE stock solution was added to the flask,and the batch test was initiated by placing the flask in a water bath shaker at 25 \({}^{\circ}\)C and 120 rpm.  
Samples were collected after 1, 3, 5, 10, 15, 30, 60, 180, 380, and 1440 min of contact time. The suspensions were then filtered through 0.45 \(\upmu\)m syringe-driven filters (LMS Co., Ltd., Hongo, Bunkyo-ku, Tokyo, Japan), and the residual concentrations of Nd, Y, Ce, and Er were analyzed using ICP-AES (ICPE-9820, Shimadzu Corporation, Kyoto, Japan) (margin of error = \(\pm\)2%).  
To determine the equilibrium time and concentration of the REEs on the resin, the concentration of the aqueous phase was plotted against contact time, and the slope of the resulting line was used to calculate the rate constant. The data were fitted with the pseudo-first-order and pseudo-second-order kinetic rate law models. It is crucial to note the assumptions of both models, as follows: (1) adsorption occurs at specific sites, where no interaction occurs between the solutes adsorb; (2) the adsorption energy is independent of the surface coverage; (3) attainment of monolayer coverage on the adsorbent surface yields maximum adsorption; (4) the concentration of the solutes does not change; and (5) adsorption of the solutes is controlled by either pseudo-first-order and pseudo-second-order rate equation [36].  
The pseudo-first-order can be expressed as:  
\[q_{t}=q_{e}\Big{(}1-e^{-k_{1}t}\Big{)} \tag{1}\]  
where:  
\(k_{1}\)--is the rate constant of the pseudo-first-order adsorption  
\(q_{e}\)--is the amount of solute adsorb at equilibrium  
\(q_{t}\)--is the amount of solute adsorb at time t  
\(t\)--is the contact time  
The pseudo-second-order model can be expressed as:  
\[q_{t}=(k_{2}\times{q_{e}}^{2})/(1+k_{2}\times t)\]  
where:  
\(k_{2}\)--is the rate constant of the pseudo-second-order adsorption  
\(q_{e}\)--is the amount of solute adsorb at equilibrium  
\(q_{t}\)--is the amount of solute adsorb at time t  
\(t\)--is the contact time  
The pseudo-second-order model can be rearranged into a linear form:  
\[t/q_{t}=1/(k_{2}\times{q_{e}}^{2})+t/q_{t} \tag{2}\]  
The equilibrium time was determined from the intersection of the equilibrium line with the time axis, which represented the time when the concentration of REEs on the resin reached a steady state. To ensure reproducibility and reliability of the results, the experiments at 3, 5, 10, and 15 min of all REEs were replicated three times.