Terbium Ion Adsorption from Aqueous Solution by Using Magnetic γ-Fe2O3-NH4OH@SiO2 Nanoparticles Functionalized with Amino Groups

New magnetic stabilized and functionalized core@shell nanoparticles (NPs) were synthesized in a simple way and characterized in order to adsorb Tb3+ from aqueous solution with a very low Tb3+ concentration. For the fluorescence determination of adsorption efficiency and capacity, tiron monohydrate as a ligand was used. The obtained results confirm the potential of the synthesized magnetic γ-Fe2O3-NH4OH@SiO2 NPs, functionalized with (3-Aminopropyl) trimethoxysilane (APTMS), to be used for adsorption of Tb3+ from aqueous solution, with the possibility of its removal from aqueous solution via an external magnet. The endothermic and spontaneous adsorption follows a pseudo-second-order kinetic model, and the adsorption equilibrium data fit the Temkin isotherm well. The maximum adsorption efficiency from aqueous solution with a 2 × 10−6 M concentration of Tb3+ is over 90% at pH 7.


Introduction
Rare earth elements (REEs) are used as components in the energy and transport sectors, environmental protection, digital technology, and medical applications in order to reduce energy consumption, improve mobility and reduce weight, reduce emissions, improve lifestyle and communication, improve medical therapy, and improve life [1][2][3][4]. Due to their low natural abundance and high production demands, Nd, Y, Eu, Tb, and Dy are considered to be the most economically critical REEs [5]. The balance between the natural abundances of the REEs in ores and the economic market demand is a major balance problem for producers of these elements [6,7]. The main source of REEs is China, which has abundant REEs reserves and advanced processing technologies [7][8][9]. In order to solve the balance problem as well as to reduce the dependence on China, the recycling market for REEs has great potential due to the increasing consumption of REE-containing products. As REEs are important for many electronic products, many nations have started looking for alternative supplies of these elements. One potential source of REEs-containing waste is scrapped electronics. In E.U. countries, electronic waste could in theory cover a significant part of the demand for REEs. Furthermore, recycling of REEs should be an important part of a circular economy, especially considering the large resource demands and negative environmental impact of current REEs production processes.
The REEs recycling methods include pyro-metallurgical, electrochemical, and hydrometallurgical approaches [10,11]. Hydrometallurgical methods, such as precipitation, solvent extraction, were analyzed by using nonlinear methods, and the thermodynamic parameters were derived from the obtained experimental results.

Characterization of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs
The synthesized γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs were characterized in order to perform a structural study, thermogravimetric analysis, morphological and compositional study, and surface area and charge study, and to determine the magnetic properties.

Structural Study
The presence of the magnetic γ-Fe 2 O 3 NP core of the particles was confirmed by X-ray diffraction (XRD). All diffraction peaks in Figure 2 are consistent with the standard structure of maghemite (JCPDS Card No.: . By taking three of the most intensive peaks, corresponding to the crystal planes of (311), (440), and (511), the average crystalline size of the γ-Fe 2 O 3 NPs is (19.1 ± 3.3) nm, as calculated by the Debye-Scherrer's equation. FT-IR spectra of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs are presented in Figure 3. The characteristic peak at 3362 cm −1 indicates the stretching vibrations of O-H groups; the peaks at 2934 cm −1 and 2890 cm −1 correspond to the stretching vibration of -CH 2 -NH 2 groups. The peak at 1635 cm −1 can be assigned to the OH groups. The peak at 1615 cm −1 indicates the scissoring vibrations of N-H in NH 2 groups; meanwhile, the peak at 1449 cm −1 confirms the presence of NH 4 + groups. The peak at 1336 cm −1 is significant for the wagging and twisting vibration of the -CH 2 -NH 2 bond. The asymmetric stretching vibration of the Si-O-Si bond corresponds to the peak at 1038.4 cm −1 . The absorption peak at 781 cm −1 corresponds to wagging and twisting of NH 2 amino groups, and the absorption peak at 562.9 cm −1 corresponds to stretching vibrations of the Fe-O bond. Figure 3 shows the shifting of peaks, corresponding to characteristic groups of synthesized γ-Fe 2 O 3 -NH 4   The FT-IR spectra of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs after adsorption of Tb 3+ exhibit shifts, intensity variations, and broadening of the peaks. With respect to the investigation of Dubey and Gandhi [34], these modifications in FT-IR spectra confirm the presence of contact between the NPs and Tb 3+ , as shown in Figure 3.

Thermogravimetric Analysis
The results of thermogravimetric analysis (TGA) are shown in Figure 4, where the mass loss of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs by heating up to 800 • C is presented. The TGA curve shows that the weight loss occurs in two steps. The first step, probably related to the evaporation of absorbed moisture, can be observed in the range from 25 • C to 180 • C and leads to a loss of 4.7%. The second decomposition step occurs in the interval from 280 • C to 800 • C after a plateau where the material is thermally stable. This step corresponds to a weight loss of 12.1% and is associated with thermal destruction of amino NH 2 groups and removal of the alkyl chains from the Si-O-Si framework (cracking of the residual surface alkyl siloxane Si-O-Si species) [36][37][38].

Morphological and Compositional Study
The TEM photos in Figure 5 display the morphology of the γ-Fe 2 O 3 NPs and γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs. Figure 5a shows the relatively spherical morphology of the maghemite NPs with the narrow particle size distribution of (13.7 ± 4.1) nm. The TEM images in Figure 5b show the relatively spherical morphological structure of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs samples with an eminent surface layer of amorphous siloxane (SiO 2 ). The average size of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs is about 55 nm. The electron diffraction pattern of the γ-Fe 2 O 3 sample in Figure 5c confirms the formation of a spinel crystal structure with a relatively high material crystallinity, which is consistent with the HRTEM image of γ-Fe 2 O 3 NPs in Figure 5d.  Note that the peaks of Cu and C in the spectra belong to the TEM's copper-grid-supported transparent carbon foil; the sample was measured on this grid during the EDXS analysis [39,40]. From the obtained EDXS pattern, it can be concluded that the amino groups of the APTMS are successfully attached to the surface of the γ-Fe 2 O 3 -NH 4 OH@SiO 2 , which is in a good agreement with the ζ-potential, as shown in Figure 7.

Surface Area and Charge Study
Brunauer-Emmet-Teller (BET) analysis gives a specific surface area of 6.41 m 2 /g for γ-Fe 2 O 3 -(NH 4 OH)@SiO 2 (APTMS) NPs. At Barrett-Joyner-Halenda (BJH) adsorption, the average pore size is 5.4 nm with a total pore volume of 0.0505 cm 3 /g. Meanwhile, at BJH desorption, the average pore size is 5.7 nm with a total pore volume of 0.05309 cm 3 /g. According to the BET specific surface area at a relative pressure of 0.3, the predicted average γ-Fe 2 O 3 -(NH 4 OH)@SiO 2 (APTMS) NP size is 53.6 nm.
In Figure 7, the ζ-potential as a function of pH of the synthesized γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs is shown. It can be seen that the isoelectric point is equal to 9.5, which confirms functionalization with amino groups from APTMS. The magnetic stabilized and functionalized core@shell NPs already show stability at pH 7, while they are fully stable at pH < 4.5, where the ζ-potential is >30 mV, and at pH > 12, where the ζ-potential is <−30 mV. Figure 8 displays the magnetization curve of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs at room temperature, where the M S value is 30.94 emu/g. However, this value of M S is similar to that of maghemite NPs [41,42]. These NPs show close superparamagnetic behavior with smaller remanence magnetization and smaller coercitivity than γ-Fe 2 O 3 NPs. The obtained magnetization properties confirm simple separation of Tb 3+ /γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs from aqueous solution.

Adsorption Study
The absorption wavelength of the prepared standard solutions, obtained by UV/VIS spectroscopy, is 312 nm. The calibration curves of the prepared standard solutions with a low concentration of Tb 3+ , obtained by a Perkin Elmer LS 55 fluorescence spectrometer, are given in Figure 9. The correlation coefficient of the calibration curves is higher than 0.99.

NP Mass Influence
The influence of the mass of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs on the adsorption efficiency is presented in Figure 10. The equilibrium mass m NPs,eq of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs as adsorbent for Tb 3+ , with respect to the adsorption efficiency, was 30 mg for a solution volume V = 20 mL, t ad = 120 min, c ads,0 = 2 × 10 −6 M, and T ad = 25 • C, where the reached adsorption efficiency was 93%. An ultrasonic bath was used to mix the samples at pH 7. Figure 10 shows that the adsorption efficiency increased significantly by increasing the mass of NPs up to 30 mg due to an increase in the number of adsorption sites. Further increasing of the mass of NPs does not influence the adsorption efficiency; therefore, 30 mg of NPs was taken as m NPs,eq (i.e., an adsorbent dosage of 1.5 mg/mL) for further investigation of the adsorption process. This value offers a good compromise between cost and efficiency.

Kinetic Study
The adsorption kinetics of Tb 3+ by γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs is shown in Figure 11. One can see that Tb 3+ were completely removed from the aqueous solution within 120 min of contact time (V = 20 mL, m NPs = 30 mg, c ads,0 = 2 × 10 −6 M, and T ad = 25 • C). Among the tested models (pseudo-first order, Elovich, and pseudo-second order), the experimental data fitted very well with the pseudo-second order rate equation with an R 2 value close to 0.9995. The appropriate values of parameters of the pseudo-second order kinetic model are as follows: k 2 = 0.3030 g·mg −1 ·min −1 and q ad,e = 0.2041 mg/g for Tb 3+ adsorption. The rate-limiting step may be chemical adsorption or chemisorption through sharing or exchange of electrons between adsorbents and adsorbates. In the initial time interval of 120 min, the adsorption efficiency increases rapidly, which means that the chemical adsorption worked at a low rate. Figure 11. The influence of time on adsorption efficiency (a) and kinetic models for Tb 3+ adsorption: pseudo-first order (b), Elovich (c), and pseudo-second order (d).

Adsorption Isotherms
The results in Figure 12 show that the adsorption efficiency increased with an increasing initial concentration of Tb 3+ c ads,0 and reached a plateau at a higher concentration due to saturation of the number of binding sites in a fixed amount of NPs. The optimum initial concentration relates mainly to the availability of active sites, which is interrelated to the presence of surface functional groups. The adsorption capacity in equilibrium q ads,e for Tb 3+ with respect to the equilibrium mass concentration of adsorbate γ ads,e shows an S-type adsorption isotherm ( Figure 12). This means low adsorption at a very low Tb 3+ concentration and then a rapid increase of adsorption with the increasing Tb 3+ concentration, until there is no more space on the surface of the nanoparticle to hold the adsorbates. The S-type isotherm confirms that the synthesized NPs are a porous material with the cohesive forces between adsorbate and adsorbent being greater than those between adsorbate molecules. To characterize the adsorption equilibria, the linear data from Langmuir, Freundlich, Dubinin-Radushkevich, and Temkin isotherms for Tb 3+ as adsorbate are presented in Figure 12. The experimental data fitted very well with the Temkin adsorption isotherm with an R 2 value of 0.9333 for Tb 3+ . The values of appropriate parameters of the Temkin isotherm are as follows: b T = 1.1605 × 10 4 J·g·mol −1 ·mg −1 and A T = 8.4242 × 10 4 L/g for Tb 3+ . The Temkin isotherms confirm the chemisorption and the assumptions that the adsorption energy decreases linearly with the increase in coverage of the adsorbent surface and that the binding energy is uniformly distributed. Due to the fact that the Temkin isotherm model fits very well with the Tb 3+ adsorption data, it can be concluded that the heat of adsorption of all molecules of the tested Tb 3+ in the γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NP layer is not the same. Furthermore, the adsorption is not monolayer, and transmigration of adsorbate in the plane of the NP surface is present.
The maximum adsorption capacity relates mainly to the availability of an active site, which is correlated to the presence of surface functional groups. The neutral nitrogen atoms of -NH 2 and -NH-groups have a lone pair of electrons that could be shared with rare earth ions, thus leading to the formation of a coordination bond between the N atoms of the amino groups from APTMS and rare earth atoms. The coordination number of Tb 3+ is 9. Due to the hypothesis that the primary and secondary amino groups are the most important sites for Tb 3+ adsorption, the predicted adsorption mechanism is based on coordination of Tb 3+ with one or two amino groups. Meanwhile, the remaining coordination sites may be provided by water molecules [43]. The predicted adsorption mechanism for Tb 3+ adsorption on the surface of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs can be written as:

Thermodynamic Study
The Tb 3+ adsorption efficiency on γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs at various temperatures and the linearization of the Van't Hoff equation are presented in Figure 13. The thermodynamic data are given in Table 1.  In the case of Tb 3+ adsorption on γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs, a change in the standard enthalpy or the adsorption heat ∆H 0 > 0 is obtained. Therefore, this adsorption is an endothermic reaction. The resulting positive change in the standard entropy ∆S 0 > 0, similar to the negative Gibbs free energy ∆G 0 < 0, indicates a spontaneous adsorption reaction. The endothermic and spontaneous adsorption reaction of Tb 3+ on γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs can be explained by good dissolution of Tb 3+ in water. When dissolving Tb-salt in water, water molecules coat the ions, and molecular bonds between the ions and water are formed: hydration. In order for Tb 3+ to be adsorbed, they must lose a part of their hydrating coat. This ion dehydration process requires energy that is greater than the released heat during adsorption of the ions on the surface of the adsorbent. Therefore, the removal of water from the Tb 3+ coating is an endothermic process whose energy exceeds the energy of the exothermic reaction of adsorption of ions that are attached to the surface. The obtained Gibbs energy values are negative, which is expected for a spontaneous adsorption process.

Materials
For each used material, its chemical formula, purity, molar mass, density at 25 • C, and Chemical Abstract Service (CAS) number and supplier are given as follows: 2-propanol (C 3 3 , 121.14 g/mol, 0.8 g/mL, CAS 77-86-1, Merck). All chemicals were used as received without further purification. For the preparation of all suspensions and solutions, deionized water (dH 2 O) was used.

Synthesis of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs
At first, 25% NH 4 OH (150 mL) was heated up to (87 ± 2) • C and stirred (300 rpm). The solution (50 mL of 0.5 M) of FeCl 2 ·4H 2 O (1.6735 g) and FeCl 3 ·6H 2 O (4.5505 g) was added to the prepared ammonium solution and heated by reflux at (87 ± 2) • C. The reaction (60 min, pH 9.9) was carried out in a round-bottom reaction flask. The particles were precipitated and separated from the reaction mixture by a permanent magnet and washed several times with dH 2 O. The washing process includes addition of dH 2 O, decantation of NPs by a magnet, and removal of the supernatant. The 25% NH 4 OH solution (25 mL) was added to the washing sediment of γ-Fe 2 O 3 NPs (~1.6 g) into the reaction flask. After the reaction (24 h, 50 • C) under continuous stirring, the stabilized γ-Fe 2 O 3 -NH 4 OH NPs were separated from the supernatant with a permanent external magnet and the dH 2 O was added. The silica shell from TEOS and functionalization with APTMS was performed "in-situ". For this purpose, 2-propanol (66 mL), dH 2 O (15.42 mL), 25% NH 4 OH solution (1.7 mL), the prepared γ-Fe 2 O 3 -NH 4 OH colloid suspension (4.93 mL, 1.038 g/mL), TEOS (0.324 mL), and APTMS (0.518 mL) were mixed under magnetic stirring (500 rpm) in a closed vessel (24 h, 25 • C). After the completed reaction, the sediment was washed with ethanol and dH 2 O several times, and the magnetic stabilized and functionalized γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) core@shell NPs were decanted by using an external magnet.

Characterization and Adsorption Techniques
The synthesized γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs were characterized in order to analyze their structural, morphological, thermogravimetric, surface charge, and magnetic properties. XRD analysis was used for structural characterization with a Brucker D4 Endavor X-ray diffractometer, equipped with Cu Kα radiation (λ = 0.15418 nm); the measurement was done at room temperature within the range of Bragg's angle 2θ from 20 • to 80 • . A Fourier Transform Infrared (FT-IR) spectrometer of the type PerkinElmer with Spectrum GX software was applied to qualitatively identify the functional groups on the material. FT-IR spectra were recorded at room temperature in a range of 400-4000 cm −1 in the transmittance mode. The amounts of the functional groups were quantitatively determined via a potentiometric titration using a two-burette Mettler T-70 instrument, equipped with a Mettler T DG 117 combined glass electrode. The burettes were filled with 0.1 M HCl and 0.1 M KOH. The stability criterion for taking a reading after each addition was set to 0.1 mV/20 s, where 20 s was the minimum time to reach equilibrium conditions between additions of the titrant, and the maximum time was set to 180 s. Thermogravimetric analysis (TGA) was performed on the dried powdered samples (previously dried at 80 • C for 10 h) in alumina crucibles, with the heating rate of 20 • C/min by using a PerkinElmer TGA 4000 under a nitrogen atmosphere (20 mL/min) up to 900 • C. The particles were observed by transmission electron microscopy (TEM) using a Jeol JEM-2100 (JEOL USA, Inc., Peabody, MA, USA), where the nanoparticles were deposited on a copper-grid-supported, perforated, and transparent carbon film. Energy Dispersive X-ray Spectroscopy (EDXS) mapping analysis was performed at 200 kV using a JEOL-2010 microscope. The Brunauer-Emmet-Teller (BET) technique was used to determine the specific surface area of nanoparticles by using a Micromeritics Flow Prep 060 with a Tristar II 3020. All samples were degassed at 110 • C for 24 h prior to each measurement. The specific surface area was evaluated in the 0.05-0.3 range of relative pressure in nitrogen gas at a temperature of 77.35 K after 24 h. To determine the electro kinetic effects at a moving interface, Dynamic Light Scattering (DLS) analysis was used, where the ζ-potential was measured by ZetaSizer Nano series Malvern Instruments. The saturation magnetization of NPs was studied by a Vibrating Sample Magnetometer (VSM, Lake Shore 7400) at a magnetic field H of 10,000 Oe and room temperature. The excitation wavelength of fluorescence spectroscopy was determined by a Perkin Elmer 35 UV/VIS spectrophotometer. To determine the amount of adsorbed Tb 3+ on the surfaces of NPs, the emission wavelengths were identified using a Perkin Elmer LS 55 fluorescence spectrometer. In order to increase the intensity of Tb 3+ fluorescence, complexes of Tb 3+ with tiron monohydrate as a ligand were used. The calibration curve was determined with standard solutions, prepared from 10 −4 M Tb 3+ solution, 10 −4 M C 6 H 4 Na 2 O 8 S 2 ·H 2 O solution, and 0.05 M tris(hydroxymethyl) aminomethane solution at a pH of 7 and a temperature of 25 • C. The measurements of fluorescence of Tb 3+ complexes were performed with sample solutions, where, instead of 10 −4 M Tb 3+ solution, the same volume of supernatant (after removal of adsorbed Tb 3+ on NPs) was used.

Procedure for Adsorption
The adsorption study was performed by mixing the prescribed mass of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs (m NPs ) with a water solution of initial Tb 3+ concentrations (c ads,0 ) at a prescribed temperature (T ad ) and at a prescribed adsorption time (t ad ). The adsorption capacity of the mass (mg) of adsorbed Tb 3+ per mass (g) of NPs q ad (mg/g) and adsorption efficiency q' ad (%) were calculated by using Equations (2) where c ads,0 (mol/L) and c ads,e (mol/L) are the initial and equilibrium concentrations of Tb 3+ , respectively, V (L) denotes the solution volume, M ads (g/mol) is the molar mass of the adsorbate, and m NPs (g) is the mass of NPs.
To determine the affinity of Tb 3+ on the γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NP surface, the adsorption conditions (equilibrium mass of adsorbent, adsorption time, initial Tb 3+ concentration, and temperature of adsorption) were changed. The Tb 3+ /γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs were removed from aqueous solution with an external permanent magnet. For determination of the adsorption capacity, UV/VIS spectroscopy and fluorescence spectroscopy were used.
The influence of the equilibrium mass m NPs,eq of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs as an adsorbent up to 40 mg (to weigh a mass, the Mettler Toledo XPE205 balance was used) for Tb 3+ on the adsorption was tested for: V = 20 mL, t ad = 120 min, c ads,0 = 2 × 10 −6 M, and T ad = 25 • C.
Then, the influence of adsorption time (up to 1440 min) at the determined m NPs,eq for Tb 3+ , V = 20 mL, c ads,0 = 2 × 10 −6 M, and T ad = 25 • C was investigated. The equivalence adsorption time t ad,eq and the most appropriate kinetic model were determined. The tested kinetic models were pseudo-first order, Elovich, and pseudo-second order models using the equations shown in Table 2, where the symbols used are: q ad,t (mg/g), adsorption capacity in time t; q ad,e (mg·g −1 ·min −1 ), equilibrium adsorption capacity; t (min), adsorption time; q ad,e,exp (mg/g), experimentally obtained equilibrium adsorption capacity; k 1 min −1 , a constant of the pseudo-first order model; α E (mg·g −1 ·s −1 ), the initial sorption rate in the Elovich model; β E (g/mg), a constant related to the extent of surface coverage and activation energy for chemisorption in the Elovich model; and k 2 (g·mg −1 ·min −1 ), a constant of the pseudo-second order model [44,45]. In the pseudo-first order kinetic model, the appearance of adsorption is assumed only on localized sites. Further assumptions are: no interaction between adsorbate ions, independence of adsorption energy on surface coverage, and the influence of the saturated monolayer of adsorbates on the NP surface on maximum adsorption. On the other hand, the Elovich model considers the interaction between adsorbate ions by adsorption on localized sites of NPs and a linear increase of the adsorption energy with the surface coverage. The pseudo-second order kinetic model is based on the assumption that the rate-limiting step may be chemical adsorption involving valence forces through sharing or exchange of electrons between the adsorbent and adsorbate, where the removal from a solution is due to the physicochemical interaction between two phases. Table 2. Kinetic models for adsorption.

Kinetic Model Nonlinear Form Linear Form Linear Plot
Pseudo-first order q ad,t = q ad,e · 1 − e −k1·t ln q ad,e − q ad,t = −k 1 t + ln q ad,e ln q ad,e,exp − q ad,t = f (t) Elovich The initial Tb 3+ concentration was varied (up to 4 × 10 −6 M) in order to study the adsorption isotherm of Tb 3+ on the surface of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs at V = 20 mL, m NPs,eq , t ad,eq , and T ad = 25 • C. Low Tb 3+ concentrations were used due to investigation of the adsorption of Tb 3+ from aqueous solution with a very low concentration of Tb 3+ (from 5 × 10 −7 M to 4 × 10 −6 M). The experimental data were interpreted by Langmuir, Freundlich, Dubini-Radushkevich, and Temkin adsorption isotherms using the equations shown in Table 3, where the symbols used are: q ad,e (mg/g), equilibrium adsorption capacity; q m (mg/g), adsorption monolayer coverage capacity; K L (L/mg), Langmuir isotherm constant; γ ads,e (mg/L), equilibrium mass concentration of adsorbate; K F ((mg/g)·(L/mg) 1/n ), Freundlich isotherm constant; n(/), dsorption intensivity; q s (mg/g), theoretical isotherm saturation capacity; K DR (mol 2 ·kJ −2 ), ε DR (kJ 2 ·mol −2 ), Dubinin-Radushkevich constant; T (K), temperature; R (8.314 J·mol −1 ·K −1 ), a gas constant; b T (J·g·mol −1 ·mg −1 ), Temkin isotherm constant; A T (L/g), the Temkin isotherm equilibrium binding constant [46][47][48][49][50]. The Langmuir isotherm describes quantitatively the formation of a monolayer adsorbate on the outer surface of NPs, and after that no further adsorption takes place. The Langmuir isotherm is valid for monolayer adsorption onto a surface containing a finite number of identical sites. The model assumes uniform energies of adsorption onto the surface and no transmigration of the adsorbate in the plane of the surface. The Freundlich isotherm describes the adsorption characteristics of a heterogeneous surface. The Dubinin-Radushkevich isotherm model expresses the adsorption mechanism with a Gaussian energy distribution onto a heterogeneous surface of NPs. The Temkin isotherm model assumes that the heat of adsorption of all molecules in the layer decreases linearly rather than logarithmically with coverage, and it takes into account the adsorbate-adsorbent interaction. ads,e log q ad,e = 1 n log γ ads,e + logK F log q ad,e = f log γ ads,e Dubinin-Radushkevich q ad,e = q s · exp ·K DR ·ε 2 DR , ε DR = R·T· ln 1 + 1 γads,e ln q ad,e = ln q s − K DR ·ε 2 DR ln q ad,e = f ε 2 DR Temkin q ad,e = R·T bT · ln A T γ ads,e q ad,e = RT bT ln γ ads,e + RT bT ln A T q ad,e = f ln γ ads,e The thermodynamic analysis of Tb 3+ adsorption on the surface of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs was done at various temperatures of adsorption. The free energy ∆G 0 was evaluated by applying Gibb's free energy (Equation (4)) and the standard enthalpy change ∆H 0 ; entropy change ∆S 0 was calculated from the Van't Hoff Equation (5): q ad,e γ ads,e (4) where ∆G 0 (J/mol) denotes the Gibb's energy change, ∆H 0 (J/mol) is the standard enthalpy change, ∆S 0 (J·mol −1 ·K −1 ) is the entropy change, T (K) is the adsorption temperature, R (8.314 J·mol −1 ·K −1 ) is the gas constant, k d (/) is the distribution coefficient, γ r (1000 g/L) is the mass concentration of water, q ad,e (mg/g) is the equilibrium mass concentration, and γ ads,e (mg/L) is the equilibrium mass concentration of the adsorbate in aqueous solution.
On the basis of the calculated distribution constant K d and Gibb's energy ∆G 0 at various temperatures and from the linearization of Equation (3) by a linear plot of ∆G 0 = f (T), the standard enthalpy change ∆H 0 and the entropy change ∆S 0 were determined.

Conclusions
Magnetic γ-Fe 2 O 3 NPs stabilized with NH 4 OH, coated by SiO 2 , and functionalized by APTMS were synthesized, characterized, and investigated as an adsorbent for Tb 3+ adsorption from aqueous solution. One of the advantages of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs is their simple synthesis procedure and good properties, confirmed by XRD, FT-IR, TGA, TEM, EDXS, DLS, and VSM characterization and the adsorption process. The experimental data of endothermic spontaneous adsorption of Tb 3+ were well-fitted to the Temkin's adsorption isotherm and pseudo-second order kinetic model. The grafted and functionalized NPs have good magnetization properties. Therefore, NPs with adsorbed Tb 3+ can be easily retrieved from the solution using a small external magnet. The main advantage of this study is that adsorption takes place at a neutral pH, which is environmentally friendly. Furthermore, the obtained high adsorption efficiency of the low concentration of Tb 3+ from aqueous solution confirms a promising application of γ-Fe 2 O 3 -NH 4 OH@SiO 2 (APTMS) NPs as an adsorbent for Tb 3+ . The maximum adsorption efficiency was obtained at a contact time of 120 min, an adsorbent dosage of 1.5 mg/mL, an initial concentration of Tb 3+ of 2 × 10 −6 M, and at a temperature of 25 • C. In future work, it might be reasonable to investigate the optimal adsorption conditions (pH, agitation speed), the possibility of adsorption of other REE 3+ on these functionalized magnetic nanoparticles, as well as the separation of various REE 3+ by selective uptake.