Kinetics of Ion Exchange in Magnesium Sulfate Leaching of Rare Earths and Aluminum from Ionic Rare Earth Ores

Abstract

Magnesium sulfate leaching of ionic rare earth ores is generally characterized by a smooth outflow curve, a long leaching time, and a high impurity content in the leach liquor. To reveal the leaching law of rare earth cations and impurity aluminum ions in the leaching process of ionic rare earth ores in magnesium sulfate, equilibrium leaching and leaching kinetics experiments were carried out using ore samples of five particle sizes (<0.10, 0.10–0.25, 0.25–0.50, 0.50–1.00, and >1.00 mm). Furthermore, prediction models of equilibrium constants and rate constants were constructed based on ion-exchange theory. The results show that the equilibrium constants of the rare earth and aluminum ion-exchange reactions decrease gradually with the increase in the magnesium ion concentration, the decrease in the temperature, and the increase in the surface area of the particles. Moreover, the equilibrium constant prediction models of rare earth and aluminum with magnesium sulfate were constructed using data fitting. From the leaching kinetics experiment, there is a significant relationship between the reaction rate constant of ion exchange and the surface area of the particles: the larger the particle size, the smaller the reaction rate constant. Based on the kinetic test data and the Arrhenius equation, the frequency factors and activation energies of the ion-exchange reactions were inversely analyzed through the Chemistry Reaction Module of COMSOL. The reaction activation energy for rare earth and aluminum leaching is 10,743 J/mol and 10,987 J/mol, respectively. The rate constant prediction model was obtained by fitting the analyzed rate constant data. The rare earth and aluminum leaching results for the full-grade ores are in high agreement with the predictions of the constructed model, which verifies the validity of the proposed model. This study can provide theoretical support for the improvement of the leaching efficiency of rare earths and the optimization of the magnesium sulfate leaching process.

1. Introduction

Rare earth is a collective name for 17 elements, including scandium, yttrium, and the lanthanides. Excluding scandium, which is widely distributed in nature, and promethium, which is radioactive, in industry, rare earths generally refer to the remaining 15 elements. According to the solubility of rare earth sulfates, rare earth elements are categorized as light rare earths (La, Ce, Pr, Nd), medium rare earths (Sm, Eu, Gd, Td, Dy), and heavy rare earths (Ho, Er, Tm, Yb, Lu, Y). Ionic rare earth ores (also known as weathered crust elution-deposited rare earth ores) have complete rare earth compositions, of which the middle and heavy rare earth reserves account for more than 80% of the world’s total rare earths [1,2,3]. Ionic rare earth ores have special metallogenic characteristics and storage forms and can be leached using ion exchange with a leaching agent. A typical rare earth resource leaching process is shown in Figure 1. Currently, ammonium sulfate is mainly used in industry as a rare earth leaching agent. However, while ammonium sulfate leaches rare earth resources, a large amount of ammonium salts remains in the ore body, which triggers the problem of ammonia and nitrogen pollution in mines and surrounding areas [4].

In recent years, researchers have attempted to use magnesium sulfate to leach rare earths. Magnesium salt is used as a leaching agent instead of ammonium sulfate. The use of magnesium salt as a leaching agent instead of ammonium sulfate has great environmental and social benefits [6,7]. However, it was found that, when magnesium sulfate leaches rare earths, the outflow curve is smoother, and the leaching time is longer. In order to improve the leaching efficiency of magnesium sulfate more effectively, an in-depth study of its leaching kinetics is needed [8].

At the same time, the selectivity of the leaching agent is poor. When rare earth ions are exchanged, impurity ions are also simultaneously leached [9]. The presence of impurity ions in the leach liquor degrades the separation efficiency between different rare earth ions and reduces the purity of the rare earth products. Studies have shown that the concentration of rare earths in the leach liquor is in the range of 500~2000 mg/L, while the concentration of the main impurity, i.e., aluminum ions, is in the range of 200~3000 mg/L [9]. To this end, the leaching kinetic model constructed not only analyzes the leaching behavior of rare earths but also describes the leaching characteristics of the aluminum impurity.

Ionic rare earth leaching is essentially a “solvent percolation–ion exchange–solute migration” cycle process. The inorganic salt-based leaching agent mainly influences the ion-exchange process under a certain injection strength. The leaching kinetics model based on ion exchange theory can effectively describe the leaching behavior of rare earths [10]. It can determine the leaching agent dosage according to the resource reserves, which is the theoretical basis for the efficient and green mining of rare earth. The exchange reaction of magnesium sulfate with rare earth ions and aluminum ions on clay minerals can be expressed by the following equations [7,11]:

$$2\left(\mathrm{R}\mathrm{E}·{\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right)+3{\mathrm{M}\mathrm{g}}^{2+}\to 3\left({\mathrm{M}\mathrm{g}·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right)+2{\mathrm{R}\mathrm{E}}^{3+}$$

(1)

$$2\left(\mathrm{A}\mathrm{l}·{\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right)+3{\mathrm{M}\mathrm{g}}^{2+}\to 3(\mathrm{M}\mathrm{g}·{\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2)+2{\mathrm{A}\mathrm{l}}^{3+}$$

(2)

There are three main types of ion-exchange models: the first type is the adsorption isothermal model, and the commonly used models of this type are Freundlich, Langmuir, etc.; the second type is the electric double-layer model, and the commonly used models of this type are Helmholtz, Gouy–Chapman, and Stern; and the third type is the mass action model, and the commonly used models of this type are Kerr, Vanselow, and Gapon [12]. Hu et al. [13] used the Kerr model to describe the ion-exchange process of liquid-phase ammonium ions exchanging solid-phase rare earth ions. The theoretical calculated values exhibit a small error when compared with the experimental values, and the formula is simple, easy to analyze, and has better application value. Qin et al. [14] and Long et al. [15] modified and extended the Kerr model and achieved good application results in leaching agent dosage determination. The existing leaching kinetic models mainly focus on the mineral particle size [16,17,18], while it is the surface or interface of mineral particles that is the main reaction medium for ion exchange, and its size has an important effect on the efficiency of ion exchange.

In this paper, leaching equilibrium experiments and leaching kinetics experiments were carried out by focusing on the particle surface area under the framework of ion-exchange model theory. Based on the mass action model, the equilibrium constant and rate constant prediction model were constructed to reveal the leaching behavior of rare earth cations (RE³⁺) and aluminum ions (Al³⁺) during the magnesium sulfate (MgSO₄) leaching process. The results show that the equilibrium constant and reaction rate constant models obtained can be used to effectively predict the leaching patterns of rare earths and aluminum under different rare earth particle surface areas, magnesium sulfate concentrations, and temperatures, providing theoretical support for improving the leaching efficiency of rare earths and optimizing the leaching process of magnesium sulfate.

2. Method

2.1. Ore Sample

The test ore sample was taken from a weathered crust elution-deposited rare earth ore in Longnan, Jiangxi Province of China. X-ray fluorescence spectrometry (XRF, Axios Max, Malvern Panalytical Ltd., Malvern, UK) and inductively coupled plasma optical emission spectrometry (ICP-OES, iCAP 7000, Thermo Fisher Inc., Waltham, MA, USA) were used to analyze the main chemical compositions and rare earth partitioning of the sample. The results are shown in

Table 1. The main chemical composition of the ore sample (%).

Composition	SiO2	Al2O3	K2O	P2O5	Fe2O3	TiO	MgO	MnO	CaO	TREO
Content	45.802	29.863	4.575	2.276	3.388	0.523	0.247	0.091	0.035	0.096

and

Table 2. The partitioning of ion-exchangeable rare earth (%).

Composition	La	Ce	Pr	Nd	Sm	Eu	Gd	Tb	Dy	Ho	Er	Tm	Yb	Lu	Y
Content	14.59	0.79	10.06	13.43	2.93	0.41	4.63	0.74	6.13	0.97	3.57	0.36	3.41	0.30	37.68

.

As shown in Table 1, the main chemical components in the ore sample are SiO₂ and Al₂O₃, and their contents are 45.802% and 29.863%, respectively. The total content of rare earth oxide is 0.096%.

As shown in Table 2, the content of light rare earths such as lanthanum, cerium, praseodymium, and neodymium is 38.86%, and the remaining content of medium and heavy rare earths is 61.14%.

Furthermore, the ore samples were sieved using a high-frequency vibrating sieve machine (GZS-1, Qingda Instrument Co., Ltd., Tianjin, China) with apertures of 1.00 mm, 0.50 mm, 0.25 mm, and 0.10 mm, and a chassis from top to bottom. The sieved rare earth samples were baked in an electrically heated constant-temperature blast drying oven (DHG-9240A, Hongguan Instrument Co., Ltd., Shanghai, China) at 110 °C for 24 h to remove the water in the mineral samples. The rare earth ore samples were homogeneously mixed using the shifting cone method. They were then partitioned and labeled to obtain the graded rare earth ore samples. The specific surface area of the sieved samples was then tested using a specific surface area analyzer for solid particles (JW-BK300, JWGB Instrument Co., Ltd., Beijing, China) [19]. The samples were vented under vacuum at 105 °C for at least 24 h. A 1.0 g sample was taken into the analyzer and adsorbed using nitrogen. Then, the analyzer automatically performed data acquisition and analysis and outputted the sample’s specific surface area value. Meanwhile, the grade of ion-exchangeable rare earth and aluminum of sieved samples was determined by the ICP-OES (iCAP 7000, Thermo Fisher Inc., Waltham, MA, USA).

The particle size distribution of the graded rare earth ore samples and the amount of exchangeable rare earths and aluminum are shown in

Table 3. The distribution of particle size and ion-exchangeable RE₂O₃ and Al₂O₃.

Particle Size/mm	Mass Distribution/%	Specific Surface Area/(m2/kg)	RE2O3 Distribution/%	RE2O3 Grade/‰	Al2O3 Distribution/%	Al2O3 Grade/‰
<0.10	27.35	387.71	45.38	1.12	60.38	0.73
0.10–0.25	21.02	193.85	21.56	0.87	16.56	0.55
0.25–0.50	16.18	96.93	14.87	0.57	14.87	0.37
0.50–1.00	14.90	28.71	12.86	0.44	6.86	0.35
>1.00	20.54	9.6	5.33	0.14	1.33	0.10
Full-particle grade		174.32		0.83		0.61

. The proportions of particle sizes of <0.10 mm, 0.10–0.25 mm, 0.25–0.50 mm, 0.50–1.00 mm, and >1.00 mm were 27.35%, 21.02%, 16.18%, 14.90%, and 20.54%, respectively. The exchangeable rare earths and aluminum grades were calculated to be 0.83‰ and 0.61‰, respectively.

2.2. Leaching Equilibrium Experiments

Five concentration gradients of magnesium sulfate solutions of 100, 150, 200, 250, and 300 mol/m³ were taken for the dynamic equilibrium dissolution leaching test, with the reaction temperatures set at 298, 313, and 328 K, and the stirring speed set at 800 r/min, respectively [15]. The quantity of rare earth sample in each set of tests was 20 g, and the corresponding magnesium sulfate solution was 100 mL [11]. In the experiment, firstly, the rare earth samples were put into five 250 mL centrifugal bottles and injected with the corresponding concentration of magnesium sulfate. After that, the centrifugal bottles containing the samples were placed on a constant temperature magnetic stirrer (Model: 85-2A, Changzhou Yuexin Co., Ltd., Changzhou, China) for the leaching test. After 2 h of reaction, the centrifuge bottle was placed in a centrifuge (Model: TD5, Changsha Yingtai Co., Ltd., Changsha, China) to separate the solution from the precipitate at a centrifugal speed of 3000 r/min, and the supernatant was taken for the detection of the content of rare earth and aluminum ions using EDTA titration [14].

2.3. Leaching Kinetics Experiments

Three concentration gradients of magnesium sulfate solutions with concentrations of 100, 200, and 300 mol/m³ were considered for the kinetic tests, and the reaction temperatures were set at 298, 313, and 328 K. For each set of tests, 80 g of the original rare earth samples was taken, as well as 400 mL of the corresponding magnesium sulfate solutions. During the test, firstly, the rare earth sample was introduced into a three-necked bottle (volume: 500 mL) and placed in a thermostatic water bath (SHA-C, Tianjin Sedaris Co., Ltd., Tianjin, China) with a set temperature. After that, magnesium sulfate solution was injected into the three-necked bottle, and the stirrer (JJ-1A, Xicheng Xinrui Co., Ltd., Changzhou, China) was quickly started with a stirring speed of 100 r/min, and the time was calculated from this moment. After the reaction was carried out for the indicated times (0.5, 1, 1.5, 2, 3, 4, 6, 8, 10, 12, 16, 20, 25, and 30 min), the samples (2 mL/trip) [8] were collected using a pipette [20] for rare earth and aluminum ion content assays.

The leaching efficiency $L_e$ was calculated using the following equation:

$$L_e={\displaystyle \frac{C_{metal}·V_L}{G_{metal}·M}}$$

(3)

where $C_{metal}$ denote the mass concentrations of rare earth or aluminum element in leach liquor, respectively, kg/m³; $G_{metal}$ is the grade of rare earth or aluminum element in ore samples, respectively, %; $V_L$ denotes the volume of leach liquor, m³; and $M$ denotes the mass of ore sample, kg.

2.4. Test Methods for RE and Al Content

The rare earth and aluminum content in the leach liquor was determined using EDTA titration, with ascorbic acid and sulfosalicylic acid to mask impurity ions such as iron and aluminum. The pH of the solution was adjusted to 5.0–5.5, using hexamethylenetetramine as a buffer and dimethylphenol orange as an indicator, and the solution was titrated with the standard solution of EDTA from purplish red to bright yellow, which is the endpoint of the titration, and the content of rare earths therein was analyzed. After the titration of rare earths, an excess of concentrated EDTA standard solution (Overdose EDTA, O-EDTA) was added to the solution, which was kept at 90 °C in a water bath for 10 min, with the zinc standard solution back titration performed until the solution turned from bright yellow to purple-red, i.e., reached the endpoint, to analyze the aluminum content [21]. All the chemical reagents used in this study were analytically pure. All experiments were repeated 3 times, and the allowable error was ≤0.4% [22].

2.5. Theory-Based Analysis of Ion-Exchange Kinetics

Ion adsorption rare earth leaching, in essence, is a process of leachate cation adsorption, rare earth ion desorption, and ion exchange in the double electric layer on the clay surface [23,24], as shown in Figure 2. The leaching agent MgSO₄ Mg²⁺ has a strong exchange capacity. SO₄²⁻ can form a more stable coordination with rare earth ions, and rare earth and impurity cation ions on the surface of the mineral can be effectively replaced and diffused into the solution. Then, they migrate in the direction of flow under the action of percolation. The mineral particle surface or interface is the main medium for ion exchange, and its size has a great influence on the rate of ion exchange.

According to the chemical equations for rare earth/aluminum (ion) leaching (Equations (1) and (2)), the expression for the ion-exchange equilibrium constant Kc [14] is shown in Equations (4) and (5).

$$2\left(\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right)+3{\mathrm{M}\mathrm{g}}^{2+}\rightleftharpoons 3\left(\mathrm{M}\mathrm{g}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right)+2{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}$$

(4)

$$K_c={\displaystyle \frac{{\left[\mathrm{M}\mathrm{g}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right]}^3{\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]}^2}{{\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]}^2{\left[{\mathrm{M}\mathrm{g}}^{2+}\right]}^3}}$$

(5)

where $\left[\mathrm{M}\mathrm{g}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right]$denote the molar concentrations of solid-phase magnesium ions on clay mineral surface, mol/m²; $\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]$ denote the molar concentrations of liquid-phase rare earth or aluminum ions in leach liquor, respectively, mol/m³; $\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]$ denote the molar concentrations of solid-phase rare earth or aluminum ions on clay mineral surface, respectively, mol/m²; and $\left[{\mathrm{M}\mathrm{g}}^{2+}\right]$ denote the molar concentrations of liquid-phase magnesium ions in leach liquor, mol/m³.

At the initial moment, the concentration of liquid-phase rare earth ions is 0, and the content of solid-phase ammonium radical ions is 0. After the ion-exchange equilibrium, according to the principle of conservation of power, the ratio of the moles of solid-phase magnesium radical ions to the liquid-phase rare earth or aluminum ions is 3:2, and then, the following equation is obtained.

$$2S_a\left[\mathrm{M}\mathrm{g}·{\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right]=3V_L\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]$$

(6)

where $S_a$ is the particle surface area of the rare earth mineral sample, m².

After the ion-exchange equilibrium is achieved, both magnesium root ions and rare earth or aluminum ions should satisfy the law of conservation of mass, as follows:

$$V_L\left[{\mathrm{M}\mathrm{g}}^{2+}\right]+S_a\left[\mathrm{M}\mathrm{g}·{\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right]=V_L{\left[{\mathrm{M}\mathrm{g}}^{2+}\right]}_0$$

(7)

$$V_L\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]+S_a\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]=S_a{\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]}_0$$

(8)

where ${\left[{\mathrm{M}\mathrm{g}}^{2+}\right]}_0$ is the molar concentration of liquid-phase magnesium ions before leaching, mol/m³; ${\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]}_0$ is the molar concentration of solid-phase rare earth or aluminum ions before leaching, mol/m².

By reorganizing Equations (6)–(8), the molar concentrations of solid-phase magnesium radical ions, liquid-phase magnesium radical ions, and solid-phase rare earth or aluminum ions were expressed as the molar concentrations of liquid-phase rare earth ions:

$$\left[\mathrm{M}\mathrm{g}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_2\right]={\displaystyle \frac{3V_L}{2S_a}}\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]$$

(9)

$$\left[{\mathrm{M}\mathrm{g}}^{2+}\right]={\left[{\mathrm{M}\mathrm{g}}^{2+}\right]}_0-{\displaystyle \frac{3}{2}}\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]$$

(10)

$$\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]={\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]}_0-{\displaystyle \frac{V_L}{S_a}}\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]$$

(11)

Bringing Equations (9)–(11) into Equation (5) gives the relationship between the molar concentrations of rare earth and aluminum ions in the liquid phase and the equilibrium constant:

$$K_c={\displaystyle \frac{{\left(\frac{3V_L}{2S_a}\right)}^3{\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]}^5}{{\left\{{\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]}_0-\frac{V_L}{S_a}\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]\right\}}^2{\left\{{\left[{\mathrm{M}\mathrm{g}}^{2+}\right]}_0-\frac{3}{2}\left[{\mathrm{I}\mathrm{o}\mathrm{n}}^{3+}\right]\right\}}^3}}$$

(12)

The concentration of rare earth or aluminum ions at the equilibrium of the reaction [Ion³⁺] can be obtained when ${\displaystyle \frac{V_L}{S_a}}$, ${\left[{\mathrm{M}\mathrm{g}}^{2+}\right]}_0$, ${\left[\mathrm{I}\mathrm{o}\mathrm{n}{·\mathrm{C}\mathrm{l}\mathrm{a}\mathrm{y}}_3\right]}_0,$ and $K_c$ are known.

3. Results and Discussion

3.1. Equilibrium Constant of the Ion-Exchange Reactions

The data on the ionic concentrations of rare earths and aluminum in solution obtained in the leaching equilibrium experiments were extracted, and the equilibrium constants for the ion-exchange reactions in different leaching environments were calculated according to Equation (12), as shown in Figure 3 and Figure 4.

Figure 3 shows the evolution of the equilibrium constants of rare earth ores of different grain sizes concerning the concentration of magnesium sulfate, which is progressively smaller for rare earth and aluminum as the concentration of magnesium sulfate increases. The smaller the particle size of the rare earth ore particles, the larger the amount of change in the equilibrium constant. The smaller the rare earth ore particles, the larger their specific surface area, and thus, the larger the medium for ion-exchange reaction, the higher the influence of magnesium sulfate concentration. At the same time, the rare earth equilibrium constant is significantly larger than that of aluminum, which also predicts that the leaching efficiency of rare earth is significantly higher than that of aluminum.

Figure 4 shows the evolution of equilibrium constants versus magnesium sulfate concentration at different leaching temperatures, and the results show that the higher the leaching temperature, the more significant the molecular heat movement effect, and the lower the equilibrium coefficient of the ion-exchange reaction. The variability of the leaching equilibrium constants of aluminum at different temperatures is significantly greater than that of rare earth, and the leaching efficiency of aluminum is more easily disturbed by temperature.

The equilibrium constants obtained from the leaching of sieved ore samples were fitted and analyzed with MATLAB (2019b) using the lsqcurvefit function. The predictive models for the equilibrium constants of the ion-exchange reactions of rare earths and aluminum are in Equations (13) and (14):

Kc-_RE = −3.45 × 10⁻¹¹·C₀ − 2.73 × 10⁻¹¹·T + 1.03 × 10⁻⁸·ln(SS_a) + 5.10 × 10⁻⁹    R² = 0.976

(13)

Kc-_Al = 5.49 × 10⁻¹⁴·C₀² − 3.73 × 10⁻¹¹·C₀ + 7.23 × 10⁻¹³·T² − 5.39 × 10⁻¹⁰·T + 2.38 × 10⁻⁹·ln(SS_a) + 9.95 × 10⁻⁸  R² = 0.969

(14)

where Kc-_RE is the equilibrium constant of rare earth ion-exchange reaction; Kc-_Al is the equilibrium constant of aluminum ion-exchange reaction; C₀ is the initial magnesium sulfate concentration, mol/m³; T is the leaching temperature, K; and SS_a is the specific surface area of the rare earth ore particles, m²/kg.

3.2. Rate Constants of Ion-Exchange Reactions

It has been shown [16,25] that the rate constants of rare earth leaching reactions at different temperatures satisfy the Arrhenius equation, as shown in Equation (15):

$$k=A{e}^{-{\displaystyle \frac{E}{RT}}}$$

(15)

where k is the rate constant, 1/s; A is the frequency factor, 1/s; E is the reaction activation energy, J/mol; and R is the ideal gas constant, 8.314 J/mol/K.

According to the leaching kinetic experiments, the concentration curves of rare earth ions and aluminum ions in the leaching solution of rare earth samples of each particle size class were obtained under different magnesium sulfate concentrations and temperature environments. The frequency factor A and reaction activation energy E were inversely analyzed based on a perfectly mixed isothermal constant volume batch reactor system using the parameter estimation function of COMSOL software 6.0 [26]. The reaction activation energy for rare earth and aluminum leaching is 10,743 J/mol and 10,987 J/mol, respectively. Moreover, the rate constants for rare earth and aluminum leaching are shown in Figure 5 and Figure 6.

The rate constants estimated from the leaching of sieved ore samples were fitted and analyzed. The constructed prediction model for the rate constants of the ion-exchange reactions of rare earths and aluminum are shown in Equations (16) and (17):

$${K_f\text{-}}_{\mathrm{RE}}=(2.59\times {{10}^{-}}^6·{\mathit{SS}}_a+4.20\times {{10}^{-}}^7·C_0+1.15\times {{10}^{-}}^4)·{\mathrm{e}}^{\left(-{\displaystyle \frac{\mathrm{10,743}}{RT}}\right)}\hspace{1em}\hspace{1em}{\mathrm{R}}^2=0.979$$

(16)

$${K_f\text{-}}_{\mathrm{Al}}=(4.43\times {{10}^{-}}^7·{\mathit{SS}}_a+8.35\times {{10}^{-}}^8·C_0+1.07\times {{10}^{-}}^5)·{\mathrm{e}}^{\left(-{\displaystyle \frac{\mathrm{10,987}}{RT}}\right)}\hspace{1em}\hspace{1em}{\mathrm{R}}^2=0.982$$

(17)

where K_f-_RE is the rate constant of the rare earth ion-exchange reaction, 1/s, and K_f-_Al is the rate constant of the aluminum ion-exchange reaction, 1/s.

3.3. Model Validation

The equilibrium constant and rate constant models were validated by taking the leaching data of full-graded rare earth ore samples, as shown in Figure 7 and Figure 8. The experimental results are in high agreement with the predicted results, and the proposed model has a high prediction accuracy for the equilibrium and rate constants.

Then, the predicted equilibrium constants and rate constants were used to carry out simulation tests of rare earth and aluminum leaching, and the results are shown in Figure 9 and Figure 10. The test results for the concentration curves of rare earth ions and aluminum ions in the leaching solution are in high agreement with the predicted results, and the constructed prediction model can be effectively used for the prediction of the rare earth leaching law. In the aluminum ion concentration curve, a certain degree of decline in leaching after the concentration reaches the peak value is observed, and this phenomenon can be observed in the studies of He et al. [11] and Long et al. [15]. The analysis suggests that this is a competitive correlation between the reaction rate constant and the equilibrium constant for the course of the ion-exchange reaction. With an increase in the reaction rate constant, the ion concentration peaks preferentially and gradually returns to equilibrium under the constraint of the equilibrium constant.

As shown in Figure 7b, Figure 8b and Figure 9b, the temperature of 328 K results in higher reaction rate constants and lower equilibrium constants, which causes the aluminum ion concentration to peak faster at this temperature; however, the concentration is indeed lower at the reaction equilibrium. For this reason, it is important to accurately identify the competing mechanisms of reaction rate constants and equilibrium constants for rare earth and aluminum ion leaching processes and to regulate the reaction rate constants and equilibrium constants through the addition of admixtures (e.g., surfactants) [2,21,27], the adjustment of leaching agent composition (e.g., complex anions) [28,29,30], and the addition of physical fields (e.g., electric fields) [31,32]. The leaching efficiency of rare earths can be improved, and at the same time, the content of impurities in the mother liquor can be reduced.

4. Conclusions

Based on the ion-exchange model, the reversible reaction equilibrium relationship between the liquid-phase magnesium ion concentration and the solid-phase adsorbed rare earth and aluminum ion concentrations was investigated. The equilibrium constants of the rare earth and aluminum ion-exchange reactions decreased gradually with the increase in magnesium ion concentration, the decrease in temperature, and the increase in particle surface area. The equilibrium constant prediction model for the ion-exchange reaction of rare earths and aluminum with magnesium sulfate was obtained by fitting the equilibrium test data.

Kc-_RE = −3.45 × 10⁻¹¹·C₀ − 2.73 × 10⁻¹¹·T + 1.03 × 10⁻⁸·ln(SS_a) + 5.10 × 10⁻⁹

Kc-_Al = 5.49 × 10⁻¹⁴·C₀² − 3.73 × 10⁻¹¹·C₀ + 7.23 × 10⁻¹³·T² − 5.39 × 10⁻¹⁰·T + 2.38 × 10⁻⁹·ln(SS_a) + 9.95 × 10⁻⁸

From the leaching kinetic tests, there was a significant relationship between the reaction rate constant of ion exchange and the surface area of the particles, and the larger the particle size, the smaller the reaction rate constant. The Arrhenius equation for the ion-exchange reaction of rare earth and aluminum with magnesium sulfate was obtained by fitting the kinetic test data.

$${K_f\text{-}}_{\mathrm{RE}}=(2.59\times {{10}^{-}}^6·{\mathit{SS}}_a+4.20\times {{10}^{-}}^7·C_0+1.15\times {{10}^{-}}^4)·{\mathrm{e}}^{\left(-{\displaystyle \frac{\mathrm{10,743}}{RT}}\right)}$$

$${K_f\text{-}}_{\mathrm{Al}}=(4.43\times {{10}^{-}}^7·{\mathit{SS}}_a+8.35\times {{10}^{-}}^8·C_0+1.07\times {{10}^{-}}^5)·{\mathrm{e}}^{\left(-{\displaystyle \frac{\mathrm{10,987}}{RT}}\right)}$$

The model for obtaining the equilibrium constant and the reaction rate constant can be used to effectively predict the leaching law of rare earths and aluminum with different specific surface areas, magnesium sulfate concentrations, and temperatures. Additionally, the competition mechanism between ion-exchange reaction rate constant and equilibrium constant was analyzed, providing theoretical support for improving the leaching efficiency of rare earths and optimizing the leaching of magnesium sulfate.