Kinetics and Mechanism of Copper Elution from Protonated Dry Alginate Beads: Process Optimization and Stability Assessment

Abstract

Numerous studies have been conducted on the removal of heavy and toxic metals using protonated dry alginate beads (PDABs) as a cation exchanger. However, there is a scarcity of research on the kinetics of copper elution with biosorbents, despite the necessity of restoring them to their original state with undiminished biosorption capacity for reuse. This study analyzes the parameters that directly affect the elution rate of copper ions from PDABs. The parameters examined include temperature (5–80 °C), sulfuric acid concentration (0.0005–0.0153 M), stirring speed (0–500 rev min⁻¹), and different acids (HNO₃, HCl, and HClO₄). Additionally, the stability of alginate was assessed over multiple cycles. The results indicate that the elution mechanism is governed by ion exchange between copper ions and protons. The copper elution rate was significantly influenced by temperature and H₂SO₄ concentration, achieving an elution efficiency of 98.6% at 80 °C and an H₂SO₄ concentration of 0.0056 M. The kinetics of copper ion elution were adequately described by the Lagergren pseudo-first-order model. The dependence of copper elution on sulfuric acid concentration was found to be of the order of 0.4. Furthermore, intrinsic rate constants were determined, and an activation energy of 9.2 kJ mol⁻¹ was obtained within the studied temperature range. These findings indicate that copper elution is a chemically controlled process.

1. Introduction

Wastewater from industries such as metallurgy and electronics is often contaminated by toxic metals, including copper ions (Cu²⁺), which pose a threat to humans, wildlife, and other organisms [1,2]. Consequently, the effective removal of these metals is essential to ensure proper effluent quality for various applications [3], as their presence may also reduce the efficiency of biological treatments [4].

Adsorption and ion exchange are commonly used to remove heavy metals, including copper ions. Additionally, several processes are available for heavy metal removal from wastewater, such as chemical precipitation, flotation, membrane filtration, electrochemical treatments, coagulation–flocculation, and ultrasonic-assisted adsorption [5,6,7,8,9]. Nevertheless, the ion exchange process has a competitive advantage due to its low cost. Natural polymeric materials, such as alginate beads, have demonstrated high potential for metal removal due to their ion exchange properties. These beads act as size-controlled sorbents with good physical and chemical properties, high porosity, and efficient metal-binding capabilities. Their effectiveness has already been demonstrated for the removal of zinc [10], cadmium [11], and nickel [12], among others. Additionally, alginates offer advantages such as eco-friendliness, sustainability, abundant oxygen-containing functional groups, ease of modification, and widespread availability [13].

1.1. Importance of Reusing Ion Exchangers

The evaluation of an ion exchanger begins with assessing its metal removal capacity relative to the amount of exchanger used. However, the inability to desorb (elute) metals and consequently recycle the exchanger represents a major limitation for industrial applications.

Table 1. Summary of research on adsorbents since 2000 for the removal and elution of copper ions.

Resin Type	Adsorption Capacity, mg/g Resin	Acid Class and Concentration Used in Elution	Percentage Eluted, %	Decrease in Eluted Capacity	Reference
Sour orange residue	21.7	HCl, 0.1 M	99%	After the first cycle, the biosorption capacity decreased by 14%	[14]
Chelate resin	168.0	HAc-NaAc, 1.0–3.0 mol/L	100	NR	[15]
D401 resin	Only the percentage recovered was reported	HNO3, 0.5–2.5 mol/L	60	NR	[16]
Iminodiacetate chelating	1.79	HNO3 7.2 mmol/L	NR	Five sorption–desorption cycles with a small loss of adsorption capacity	[17]
Amino methylene phosphonic acid resin	181.0	HCl, 1.0–3.0 mol/L	NR	NR	[18]
Tomato waste	46.0	NR	NR	NR	[19]
Saccharomyces cerevisiae biomass	28.8	NR	NR	NR	[20]
Dried activated sludge	62.5	NR	NR	NR	[21]
Lyngbya putealis	7.8	NR	NR	NR	[22]
Caustic baker’s yeast	5.7	NR	NR	NR	[23]
Ethanol baker’s yeast	3.3	NR	NR	NR	[23]
Pristine baker’s yeast	2.4	NR	NR	NR	[23]
Pycnoporus sanguineus	2.8	NR	NR	NR	[24]
Chlorella vulgaris	58.8	NR	NR	NR	[25]
Acidosasa edulis shoot shell	2.51	NR	NR	NR	[26]
Padina sp.	72.4	NR	NR	NR	[27]
Sargassum sp.	62.9	NR	NR	NR	[27]
Ulva sp.	47.7	NR	NR	NR	[27]
Gracillaria sp.	37.5	NR	NR	NR	[27]
Peat	14.3	NR	NR	NR	[28]
Grafted silica	16.6	NR	NR	NR	[29]

NR = not reported; Hac = acetic acid; NaAC = sodium acetate.

presents various biosorbents used for copper ion removal, including studies that have evaluated their elution capacity. The studies are ranked in descending order based on the amount of information reported on elution.

Notably, only one study [14] provides comprehensive data on copper adsorption, acid type, and elution performance. Three other studies [15,16,17] report elution based on either the eluted amount or the number of reuse cycles, while the remaining 17 studies did not assess the elution capacity of their biosorbents. Overall, only 19% of copper removal studies have investigated elution. According to Alluri et al. [30], elution studies help characterize biosorption processes, including physisorption, ion exchange, and complexation. Furthermore, understanding biomass metal sorption and desorption (elution) performance is crucial for standardizing biosorption procedures for industrial use. Otherwise, biosorbents frequently exposed to acidic media with strong desorption agents, such as HCl, may suffer degradation, leading to reduced structural integrity and loss of binding sites.

The primary objective of desorption is to maintain the adsorption capacity of the biosorbent. The desorption process should allow for the recovery of metals in a concentrated form (especially in the case of economically valuable metals) while ensuring that the biosorbent retains its original biosorption capacity for reuse [30]. An ideal eluent should meet the following criteria [31]: low cost, environmental friendliness, minimal damage to the biomass, and preservation of metal-binding capacity.

Given these considerations, it is essential to comprehensively study both the elution process and the number of reuse cycles that biosorbents can undergo for industrial applications, particularly in the case of copper ion removal.

1.2. The Use of PDABs for Copper Uptake

Copper ions (Cu²⁺) can be effectively removed from highly diluted solutions using protonated dry alginate beads (PDABs). These solutions typically originate from residual streams of the solvent extraction (SX) stage. Previous studies [32] have reported a copper ion uptake capacity of 102.5 mg g⁻¹ of dry alginate at pH 5.0 after 360 min. Figure 1 provides a summary of the results obtained in this study, illustrating the effects of solution stirring (100 rev min⁻¹), pH (5.0), removal time (360 min), loading capacity (400 kg dm⁻³), and temperature (80 °C) on copper uptake. Each of these parameters represents the optimal conditions identified for copper adsorption using PDABs. The uptake mechanism was determined to be ion exchange between Cu²⁺ and protons from the functional groups of the alginate beads.

Despite these findings, no studies have been conducted on copper elution to establish the parameters affecting the desorption process from PDABs. Furthermore, the number of reuse cycles for alginate has not been determined. Notably, no bibliographic studies on copper ion elution have been reported.

This research aims to investigate the mechanisms governing copper ion elution from alginate beads under varying experimental conditions, including agitation, temperature, and sulfuric acid (H₂SO₄) concentration, as well as the use of alternative acids. Additionally, the stability of alginate will be assessed over multiple reuse cycles. Finally, a kinetic model will be developed to describe Cu²⁺ elution from PDABs, including the determination of kinetic parameters such as reaction orders and activation energy.

1.3. Chemistry and Elution Mechanism of Cu²⁺

A thermodynamic study was conducted to evaluate the solubility of copper species using electrochemical phase diagrams (Eh-pH diagrams). The Eh-pH diagram for the Cu-H₂O system was constructed at 5, 17, and 80 °C, assuming a copper concentration of 0.000362 M (23 mg L⁻¹). Thermodynamic data for this equilibrium diagram were primarily obtained from the HSC Chemistry database [33]. The copper species present in the diagram include Cu²⁺, Cu(OH)₂, Cu(OH)₂⁴⁻, Cu₂O, and Cu. Figure 2 shows the equilibrium diagram of the Cu-H₂O system at 5, 17, and 80 °C.

The diagram indicates that the predominant Cu²⁺ region, which is crucial for ion exchange, occurs at potentials above 0.24 V and under acidic pH conditions. As the temperature increases, the Cu²⁺ predominance region decreases. Specifically, for temperatures ranging from 5 to 80 °C, the equilibrium lines between Cu²⁺ and Cu(OH)₂ shift to pH values of 5.2 and 4.1, respectively. Meanwhile, the Eh ranges between Cu²⁺ and Cu remain relatively stable despite temperature variations.

The figure suggests that, within the studied pH range, Cu²⁺ remains the predominant species, making ion exchange a viable mechanism for copper removal at temperatures between 5 and 80 °C under acidic conditions. Additionally, previous studies [10,11,12] have shown that the removal of divalent metal ions (M²⁺) in PDABs can be represented by the following reaction:

2{COO-H} + M²⁺ = {COO}₂-M + 2H⁺

(1)

Therefore, the reverse reaction, which represents elution under acidic conditions, can be expressed as follows, where M²⁺ corresponds to Cu²⁺:

{COO}₂-Cu + 2H⁺ = 2{COO-H} + Cu²⁺

(2)

2. Experimental Work

Solutions with varying copper concentrations were prepared by dissolving CuSO₄·5H₂O in double-distilled water. H₂SO₄ solutions were generated using concentrated H₂SO₄ with a purity of 98%. Other acids, such as nitric acid (HNO₃), hydrochloric acid (HCl), and perchloric acid (HClO₄), also had high purity levels (above 98%). All chemical reagents were sourced from Merck Group.

Regarding PDAB generation, alginic acid chains were cross-linked with calcium (Ca) and barium (Ba) to produce PDAB-Ca and PDAB-Ba [34]. Subsequently, PDAB was protonated using a nitric acid solution, washed, and dried at room temperature. The formation of PDABs resulted in a corrugated surface with spherical beads of approximately 1.0 ± 0.1 mm in diameter and high porosity. PDABs remained physically and chemically stable within a pH range of 1 to 6; beyond this range, Ca and Ba were released from the beads, causing the PDABs to lose rigidity and stability.

2.1. Copper Ion Removal

Experiments on copper ion removal using PDABs involved contacting 80 mg of PDABs with 400 mL of a 0.000362 M copper solution for 360 min at 17.0 ± 0.1 °C. The solution pH was maintained by adding NaOH. Samples of 1 mL were collected at different intervals and analyzed using Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES). Each sample was diluted to 10 mL with double-distilled water and filtered through cellulose membrane filters. The residual concentrations of copper, barium, and calcium ions in the filtrates were analyzed using ICP-AES. At the end of the experiments, the PDABs were collected, washed with double-distilled water, air-dried, and analyzed using Scanning Electron Microscopy with Energy Dispersive Spectroscopy (SEM-EDS). To validate copper removal in the alginate, the element’s composition was analyzed using backscattered electron imaging (BEI) and quantified with an EDS Quantax system.

2.2. Copper Ion Elution

Copper ion elution experiments were conducted with 400 mL of H₂SO₄ solution at varying concentrations (0.0005 to 0.0153 M). The PDAB mass used was 80 mg. Batch-type isothermal elution experiments were performed in a 2 L glass reactor equipped with a variable mechanical stirrer, heating mantle, thermocouple, porous sampling tube, and water-cooled condenser to minimize solution loss by evaporation. The elution process began by immersing copper-loaded PDABs (80 mg) in the H₂SO₄ solution at temperatures ranging from 5 to 80 °C. Similar to the removal experiments, 1 mL samples were periodically extracted, diluted to 10 mL with double-distilled water, and analyzed for Cu, Ba, and Ca concentrations using ICP-AES. The beads were collected, washed, air-dried, and examined using SEM-EDS. Additionally, compositional imaging (BSE) was employed.

The reproducibility of both removal and elution experiments was assessed through duplicate and triplicate tests, showing variations below 2.0%. Mean values were calculated.

The percentage of copper elution was determined based on copper mass recovery from the alginate over time, as represented by Equation (3):

$$\mathrm{E}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} \mathrm{c}\mathrm{o}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r},\mathrm{\%}={\displaystyle \frac{{\mathrm{W}}_{\mathrm{C}\mathrm{u}\left(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)}}{{\mathrm{W}}_{\mathrm{C}\mathrm{u}\left(\mathrm{P}\mathrm{D}\mathrm{A}\mathrm{B}\right)}}}\times 100$$

(3)

where ${\mathrm{W}}_{\mathrm{C}\mathrm{u}\left(\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right)}$ is the mass of copper in the solution over time and ${\mathrm{W}}_{\mathrm{C}\mathrm{u}\left(\mathrm{P}\mathrm{D}\mathrm{A}\mathrm{B}\right)}$ is the initial copper mass in the PDABs.

2.3. Chemical and Solids Analysis

Chemical analysis for removal and elution experiments was conducted using ICP-AES. For morphological analysis (SEM), the samples were coated with gold to enhance resolution. Both analyses were performed using a Tescan^® Vega LSH electron microscope (Tescan, Brno-Kohoutovice, Czech Republic) equipped with a Bruker^® 6030 EDS detector (Bruker Scientific LLC, Billerica, MA, USA).

Copper concentration in the alginate beads was determined by initially identifying calcium, barium, copper, and oxygen using EDS analysis. Bruker Esprit 2 software was then employed to quantify the species and determine the relative copper concentration.

3. Results and Discussion

This section presents studies on parameters influencing copper ion elution from PDABs, including agitation rate, temperature, H₂SO₄ concentration, and the effects of various acids. First, copper extraction results from alginate analysis are presented, followed by the kinetic model representing the Cu²⁺ elution rate.

3.1. PDAB Loading Assay

Before conducting elution experiments, alginate spheres were loaded with copper ions via removal from solution. Copper removal results were obtained for an initial concentration of 23 mg L⁻¹ at pH 6.0, using 80 mg of alginate at 17.0 ± 0.1 °C. Figure 3 illustrates the decrease in copper concentration over time. The Ba and Ca concentrations were below detection limits.

A final concentration of 2.2 mg L⁻¹ was achieved, corresponding to 90.4% copper removal. This process, described in a study by Aracena [32], is attributed to an ion exchange mechanism where PDAB functional group protons exchange with copper ions (see Reaction (1), where M = Cu). Copper removal reached 97.5 mg g⁻¹ of PDABs (dry weight) after 360 min. Consequently, each elution experiment required prior alginate loading under the same conditions.

Figure 4 presents the Cu profile across alginate particles using BEI analysis.

Table 2. Concentration profile of copper on the surface of alginates through EDS analysis for a pH value of 6.0.

Point	Cu Concentration, %
	pH = 6.0
1	26.86
2	23.89
3	21.85
4	23.18

lists Cu percentages at each measurement point, showing higher concentrations at the external surface. This behavior aligns with the high copper removal efficiency from aqueous solutions.

The white fine particles dispersed in the PDABs correspond to barium and calcium, the main components that bind and keep the PDABs stable.

3.2. Effect of Agitation Rate

Copper-loaded PDABs were used for all subsequent experiments. Elution tests assessed stirring speed within 0 to 500 rev min⁻¹ using 0.4 L of a 0.0056 M H₂SO₄ solution with 80 mg of alginate at 17.0 ± 0.1 °C. The results, depicted in Figure 5, indicate that elution increased with agitation. At 150 rev min⁻¹, elution reached 47.4%, while at 500 rev min⁻¹, it was 65.2%. Beyond 400 rev min⁻¹, stirring had negligible effects on copper removal, confirming that mass transfer was not the limiting factor. Without agitation, notable elution rates of 14.3% were observed.

3.3. Effect of Temperature on Elution Rate

The application of thermal energy in ion exchange processes directly influences the removal or elution of species. In this study, experiments were conducted within a temperature range of 5 to 80 °C. The results, presented in Figure 6, demonstrate the predominant effect of temperature on the copper elution rate. At 17 °C, copper elution is significant, reaching 76.9% after 120 min, whereas at 60 °C, elution exceeds 98% within the same period. At lower temperatures, specifically 10 and 5 °C, values of 64.9% and 45.9% copper elution, respectively, were observed after 120 min, which is notable given their proximity to the freezing point of water.

The substantial differences in the elution rates of Cu from PDABs may be attributed to variations in the kinetic constants governing the elution processes.

3.4. Effect of H₂SO₄ Concentration

According to reaction (2), the presence of H₂SO₄ plays a crucial role in the Cu²⁺ elution rate. Therefore, the effect of H₂SO₄ concentration on copper elution was investigated. The results, depicted in Figure 7, indicate that at lower H₂SO₄ concentrations, copper elution is significantly slower, achieving only 13.2% at 0.0005 M H₂SO₄ after 120 min. However, increasing the acid concentration to 0.0153 M substantially enhances elution, reaching 93.8% within the same duration. Notably, at H₂SO₄ concentrations above 0.0092 M, the copper elution rate remains nearly constant, suggesting a saturation threshold beyond which further increases in acid concentration do not enhance elution. This observation is critical for determining the optimal H₂SO₄ concentration in future industrial ion exchange applications.

To analyze the surface chemical composition of PDABs after copper removal, X-ray analysis was performed on samples coated with graphite from experiments conducted at different H₂SO₄ concentrations (0.0010 M and 0.0056 M) for 120 min. The analysis, carried out using the EDS Quantax system, provided the results summarized in Figure 8 and

Table 3. Concentration of copper on the surface of alginates through EDS analysis for different concentrations of H₂SO₄.

Point	Cu Concentration, %
	Original Sample	0.0010 M	0.0056 M
1	26.86	12.08	4.90
2	23.89	19.32	8.17
3	21.85	14.39	9.01
4	23.18	13.63	9.04
Average	23.95	14.86	7.78

. The data reveal that the average copper composition on the particle surface decreases as H₂SO₄ concentration increases, aligning with the trends observed in Figure 7. Specifically, a reduction of approximately 48% in Cu content was observed when increasing the H₂SO₄ concentration from 0.0010 M to 0.0056 M.

3.5. Effect of Different Acid Reagents

In the metallurgical and chemical industries, various reagents are evaluated for their efficiency in eluting ion exchange resins. In this study, the elution performance of three commonly used reagents—HNO₃, HCl, and HClO₄—was assessed. The experimental results (Figure 9) indicate a positive effect on copper elution for HClO₄ and HNO₃. Specifically, using HClO₄ resulted in 86.5% elution after 120 min, while HNO₃ achieved a rate of 68.6% within the same period. The elution curves for these reagents closely resemble those obtained with H₂SO₄.

However, when using HCl, the elution rate was significantly lower, reaching only 18.3% after 120 min. Due to the limited literature on copper elution using these reagents, further studies are necessary to determine their effectiveness at varying concentrations.

3.6. Removal and Elution Cycle Study

The stability of alginate beads is crucial for their practical application in repeated cycles. To assess this, removal and elution tests were conducted under the same conditions as previous experiments. The removal conditions were as follows: Initial Cu concentration of 23 mg L⁻¹, pH 6.0, alginate mass of 80 mg, solution temperature of 17 °C, and an experimental duration of 360 min. The elution conditions were as follows: Initial H₂SO₄ concentration of 0.0050 M, alginate mass of 80 mg, temperature of 60 °C, and an experimental duration of 120 min. A maximum of five cycles was performed, with the results showing copper removal of 98 mg g⁻¹ and elution rates close to 90% for each cycle. These findings confirm that alginate can be reused while maintaining its structural integrity and performance. Furthermore, the beads remained physically stable without structural degradation or significant material loss. Future studies could focus on continuous systems to evaluate bead performance over extended cycles and under varying operational conditions.

3.7. Copper Elution Kinetics

As observed in the previous figures, copper elution is strongly influenced by temperature. This suggests that the elution process is controlled by chemical reactions occurring on the bead surface. To model this process, the Lagergren pseudo-first-order kinetic model [35] was applied to interpret the data and analyze the effects of temperature and H₂SO₄ concentration on elution kinetics. Additionally, a preliminary study was conducted to assess the adsorption and desorption characteristics of the same resin type. Several kinetic models were tested, including pseudo-first-order and pseudo-second-order equations, as well as intra-particle diffusion models and film diffusion mass transfer equations. Among these models, only the pseudo-first-order kinetic model successfully replicated the experimental data. The corresponding kinetic equation is as follows:

$${\displaystyle \frac{\mathrm{d}{\mathrm{q}}_{\mathrm{t}}}{\mathrm{d}\mathrm{t}}}={\mathrm{k}}_{\mathrm{a}\mathrm{p}\mathrm{p}}\left({\mathrm{q}}_{\mathrm{e}}-{\mathrm{q}}_{\mathrm{t}}\right)$$

(4)

where q_t is the eluted copper at time t, q_e is the eluted copper at equilibrium (mg g⁻¹), t is time (min), and k₁ is the apparent kinetic constant (min⁻¹). Integrating and applying boundary conditions (t = 0 at q_t = 0 and q_t = q_e at time t), Equation (4) becomes the following expression:

$$\mathrm{l}\mathrm{o}\mathrm{g}\left({\mathrm{q}}_{\mathrm{e}}-{\mathrm{q}}_{\mathrm{t}}\right)=\mathrm{l}\mathrm{o}\mathrm{g}\left({\mathrm{q}}_{\mathrm{e}}\right)-\left({\displaystyle \frac{{\mathrm{k}}_{\mathrm{a}\mathrm{p}\mathrm{p}}}{2.303}}\right)\mathrm{t}$$

(5)

k_app is the apparent kinetic constant given by:

$${\mathrm{k}}_{\mathrm{a}\mathrm{p}\mathrm{p}}={\mathrm{k}}_1{\mathrm{C}}_{{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4}^{\mathrm{p}} e^{-\left({\displaystyle \raisebox{1ex}{${\mathrm{E}}_{\mathrm{A}}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{R}\mathrm{T}$}\right.}\right)}$$

(6)

where k₁ is the linear velocity constant, ${\mathrm{C}}_{{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4}$ is the H₂SO₄ concentration, and p represents the apparent reaction order.

A graphic log(q_e − q_t) was built as a function of time was constructed for all experimental data from Figure 6 across the temperature range of 5 to 80 °C. Figure 10 illustrates the application of Equation (5).

This figure shows a strong linear fit of the kinetic data, with regression coefficients (R²) ranging from 0.986 to 0.998 across the entire temperature range, indicating the applicability of Equation (4). The apparent kinetic constants at various temperatures were obtained from the slopes of the linear fits, as presented in

Table 4. Rate constants for copper elution from the PDABs.

T [°C (K)]	1000/T (1/K)	kapp × 103	−ln kapp
5 (278)	3.5971	18.2	4.007
10 (283)	3.5336	23.9	3.732
17 (290)	3.4483	26.0	3.649
40 (313)	3.1949	34.3	3.372
60 (333)	3.0030	35.9	3.326
80 (353)	2.8329	49.5	3.005

.

The reaction order, p, was determined from the kinetic data on the effect of H₂SO₄ concentration (Figure 6). Figure 11 presents the experimental data for various H₂SO₄ concentrations plotted according to Equation (4), while the k_app values were used to generate a graph of ln k_app vs. ln ${\mathrm{C}}_{{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4}$ shown in Figure 12. This figure exhibits a linear relationship with an R² of 0.85 and a slope indicating that the reaction order with respect to the H₂SO₄ concentration is 0.4.

The intrinsic rate constants, k₁, were calculated for the studied temperature range using the apparent kinetic constants obtained from Figure 10 and a reaction order (m) of 0.4.

Table 5. Rate constants for copper elution in the temperature range of 5 to 80 °C at a H₂SO₄ concentration of 0.0056 M.

T, °C	kapp, min−1	k1, min−1 (H2SO4)−0.4
5	0.00181	0.14476
10	0.0239	0.19057
17	0.0260	0.20706
40	0.0343	0.27303
60	0.0359	0.28585
80	0.0495	0.39396

presents both the apparent and intrinsic rate constants for the temperatures analyzed. These intrinsic rate constants were then used to construct the Arrhenius plot shown in Figure 13. This figure demonstrates a strong linear fit (R² > 0.95) of the intrinsic kinetic constants at each temperature. The calculated activation energy was 9.2 kJ mol⁻¹ for the temperature range of 5 to 80 °C, which is a typical value for a surface-controlled chemical reaction. Consequently, the intrinsic kinetic constant is expressed as follows:

$${\mathrm{k}}_{\mathrm{a}\mathrm{p}\mathrm{p}}=8.72·{\left[{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\right]}^{0.4}·e\left(-{\displaystyle \frac{9.200}{\mathrm{R}\mathrm{T}}}\right)$$

(7)

where R is 8.314 J mol⁻¹ K⁻¹, [H₂SO₄] is in molarity (M), t is in minutes, and k₁ = 8.72 M^−0.4 min⁻¹.

3.8. Proposed Flowsheet for the Generation of Copper Sulfates

Figure 14 presents a proposed general flow diagram for the removal and subsequent elution of copper. Two output solutions are considered:

A highly basic solution, which can be recycled into a pool to increase the pH of the leaching solution (pH = 2.0 or lower) and continue to be used to achieve high copper removal efficiencies.

An acidic solution with a high Cu concentration, which can be further processed to generate copper sulfate (CuSO₄) crystals.

Copper sulfate pentahydrate is widely used in the agro-fisheries industry, including as an animal feed additive, a fungicide, and a cleaning agent for tanks to remove algae, among other applications.

4. Conclusions

Based on the results of this study, the following conclusions can be drawn:

• Copper elution from alginate beads occurs through an ion exchange mechanism between copper ions and protons.
• Increasing the stirring rate of the solution enhances copper ion elution. Above 400 rev min⁻¹, the amount of copper eluted is not significant.
• Higher temperatures and H₂SO₄ concentrations increase copper elution, reaching up to 98% at 80 °C and an H₂SO₄ concentration of 0.0056 M.
• Copper elution kinetics were analyzed using a pseudo-first-order kinetic model, which was employed to determine the dependence of the elution rate on temperature and H₂SO₄ concentration.
• The ion exchange reaction was found to be controlled by a surface chemical reaction, with an order of 0.4 regarding the H₂SO₄ concentration. The activation energy was calculated as 9.2 kJ mol⁻¹ for the temperature range of 5–80 °C.