Comprehensive Cost–Benefit and Statistical Analysis of Isotherm and Kinetic Models for Heavy Metal Removal in Acidic Solutions Using Weakly Base Polymeric Chelating Resin as Adsorbent

Abstract

This study investigates the removal of heavy metals, particularly copper and nickel, from acidic aqueous solutions using the weakly base polymeric chelating resin Dowex M-4195. The research conducts comprehensive cost–benefit and statistical analyses of various kinetic and isotherm adsorption models. The results show that the PSO and general order models provide high accuracy for the Cu²⁺ adsorption kinetics, while the Avrami fractional order model excels for Ni²⁺. In terms of the isotherm models, the Langmuir and Jovanovic models are highly accurate for both metals, with the Toth model being particularly effective for Ni²⁺ due to its ability to account for surface heterogeneity and multi-layer adsorption. This study also reveals that the kinetic adsorption process is more economically beneficial than the isotherm adsorption process, highlighting the importance of model selection for optimizing heavy metal removal. Incorporating circular economy principles, this research emphasizes the sustainability of using regenerable and reusable adsorbents like Dowex M-4195. The findings provide valuable insights for designing efficient adsorption systems, promoting environmental sustainability, and ensuring public health safety.

1. Introduction

Heavy metals such as copper (Cu) and nickel (Ni) are among the most persistent and toxic environmental contaminants, particularly in aquatic ecosystems [1,2]. These metals can accumulate in water bodies from a variety of sources, leading to significant ecological and human health risks. Human activities are the primary contributors to heavy metal pollution, encompassing industrial discharges, mining operations, agricultural practices, and urban runoff [3]. Industrial processes release substantial quantities of metals through effluents and emissions, while mining activities disturb the earth, exposing and releasing metals into surrounding water systems. Agricultural activities contribute to metal contamination through the use of fertilizers and pesticides, which contain trace amounts of these metals [4]. Urban runoff further exacerbates the issue by carrying metals from roads, buildings, and other infrastructure into water bodies [5]. In addition to human activities, natural processes such as volcanic eruptions, weathering of rocks, and soil leaching also contribute to the presence of heavy metals in the environment [6].

The presence of heavy metals in water poses severe risks to human health [7]. These metals are known to cause mutagenicity, endocrine disruption, carcinogenicity, genotoxicity, neurotoxicity, and skin and eye irritations [8]. Prolonged exposure to even low concentrations of heavy metals can lead to chronic health conditions and significant biological damage. As these metals do not degrade easily, they persist in the environment, accumulating in sediments and entering the food chain through aquatic organisms [9]. Fish, in particular, can bioaccumulate heavy metals, which then pose risks to humans when consumed [10].

To address the issue of heavy metal contamination in water, adsorption techniques using various adsorbents have been widely studied and employed [11,12]. Prominent among these are activated carbon, biochar, chitosan, zeolite, clay minerals, and resin [13,14,15]. Each of these materials exhibits unique properties that make it effective for this purpose. Activated carbon is widely recognized for its high surface area and porosity, which enable the adsorption of a broad spectrum of contaminants, including heavy metals. Biochar, a form of charcoal derived from biomass, offers a sustainable and cost-effective alternative, with its surface functionality and pore structure enhancing its metal-binding capabilities [16]. Chitosan, a biopolymer derived from chitin, possesses functional groups such as amino and hydroxyl groups that facilitate the chelation of metal ions. Zeolites, aluminosilicate minerals, are valued for their ion-exchange properties and high affinity for certain metals. Clay minerals, with their layered structures and large surface areas, also play a significant role in heavy metal adsorption through mechanisms such as ion exchange and surface complexation [17]. Resins, particularly ion-exchange resins, are synthetically engineered to selectively bind specific metal ions, offering high efficiency and regeneration potential [18]. In summary, the selection of an appropriate adsorbent for heavy metal removal depends on the specific application requirements, including the type of metal contaminants, the characteristics of the water matrix, and economic considerations. Each of these adsorbents offers distinct advantages that can be leveraged to optimize the removal process. One promising adsorbent is the weakly base polymeric chelating resin Dowex M-4195, which has demonstrated significant efficacy in removing heavy metals (Cu²⁺ and Ni²⁺) from aqueous solutions [19]. Dowex M-4195 is a macroporous resin with a bispicolamine functional group [20], which is capable of withstanding high temperatures and a wide pH range. Its structure allows for effective ion exchange, making it suitable for capturing and holding metal ions from contaminated water. In addition, Dowex M-4195 is an ion-exchange resin specifically designed for selective recovery of metals like copper and cobalt. It offers several advantages over its counterparts, such as higher selectivity and better performance in acidic conditions. Unlike Amberlite IR-120, Dowex M-4195 maintains its efficiency even in the presence of strong acids, making it a superior choice for applications involving harsh chemical environments [12]. Additionally, its higher affinity for specific metal ions compared to Chelex 100 allows for more efficient metal recovery processes [1].

Incorporating the principles of a circular economy is crucial for addressing heavy metal contamination sustainably. A circular economy aims to minimize waste and optimize resource use by closing product lifecycles through greater recycling and reuse [21]. In the context of heavy metal removal, the use of adsorbents like Dowex M-4195 resin aligns well with these principles. This resin can be regenerated and reused multiple times, reducing the need for constant production of new adsorbents and minimizing the environmental impact. Additionally, the metals captured by the resin can potentially be recovered and reused in industrial processes, thus closing the loop and contributing to resource efficiency [22]. The circular economy approach addresses environmental and health concerns while offering economic benefits. By recycling and reusing materials, industries can reduce the costs associated with raw material procurement and waste disposal. Furthermore, it promotes innovation in product design and process optimization, driving sustainable growth and creating new business opportunities. Implementing circular economy principles in water treatment and heavy metal removal processes enhances sustainability and resource efficiency, aligning with global efforts to transition toward a more sustainable and resilient economy [23,24].

Various research streams currently devote significant attention to the definitions and functions of various adsorbents, which are well known in the field and may not require a detailed explanation. To better align the introduction with the research, it would be more effective to briefly mention the adsorbents in relation to the current research and shift the focus toward the methodological approach, particularly the application of Akaike’s information criterion (AIC) in evaluating adsorption models. AIC is a valuable statistical tool that effectively balances the fit of a model with its complexity, making it particularly useful for comparing different adsorption models while avoiding overfitting [25]. However, some research explains why AIC was chosen over other model evaluation methods such as the Bayesian information criterion (BIC) or adjusted R-squared, highlighting AIC’s ability to optimize the trade-off between accuracy and model simplicity. Although the BIC similarly penalizes model complexity, it does so more stringently, which might not be ideal for the specific requirements of adsorption models [26]. Additionally, some research explores other methods like the BIC and cross-validation, and then clarifies why AIC is especially suitable for adsorption studies. The innovation of adsorption research lies in its application of AIC across a wider array of models and its integration with a cost–benefit analysis, offering a more thorough evaluation of models. This dual approach not only pinpoints the most statistically sound model but also considers practical applications, which is critical for enhancing the efficiency of adsorption processes in industrial settings.

Overall, this study significantly contributes to understanding the cost-effectiveness and efficiency of using polymeric chelating resins for heavy metal removal. By combining thorough statistical analysis with the principles of a circular economy, it offers a robust framework for improving water treatment processes, promoting environmental sustainability, and ensuring the health and safety of communities. The study’s methodology includes batch experiments to determine the adsorption capacity and kinetics of Dowex M-4195 for Cu²⁺ and Ni²⁺. These experiments involve varying the concentrations of heavy metals and analyzing the equilibrium adsorption using atomic absorption spectroscopy. Statistical analyses, including AIC and corrected AIC, are then applied to identify the best-fitting models. This study focuses on comprehensive cost–benefit and statistical analyses of isotherm and kinetic adsorption models for the removal of heavy metals, particularly Cu²⁺ and Ni²⁺, using Dowex M-4195 in acidic solutions. The primary objective is to evaluate and compare different adsorption models to identify the most effective ones based on Akaike’s information criterion and the corrected Akaike’s information criterion.

2. Materials and Methods

2.1. Polymeric Bispicolamine Chelating Resin Preparation

The raw precursor chelating resin Dowex-M4195 (Figure 1), featuring a bispicolamine functional group, was procured from Dow Chemical Company through Supelco. The Dowex-M4195 beads possess a macroporous structure with a size range between 3 × 10⁶ and 8 × 10⁶ angstroms. In its initial form, it is in an ionic state (Na⁺) and can withstand temperatures up to 60 °C within a pH range of 0–7. To convert the resin into its hydrogen (H⁺) form, a ratio of 1 g of chelating resin to 10 milliliters of 2 M acetic acid (AR grade, Ajax Finechem, 99.99%) was soaked together for over 24 h. This process has been detailed in prior studies [19]. Following this, the wet H⁺ Dowex-M4195 chelating resin was thoroughly rinsed with double-distilled water to remove excess acid until the pH approached neutrality. Subsequently, it was dried in an oven at 50 °C for a duration of 12 h. The resulting dried Dowex-M4195 resin in H⁺ form was carefully stored and allowed to cool to room temperature (approximately 25 °C) in a glass tube. This prepared resin was then used for further batch ion-exchange investigations (adsorption process).

2.2. Adsorption Studies

The adsorption studies via batch experiments for adsorption were operated at ambient temperature using a 250 mL closed paraffin Erlenmeyer flask. A 2000 mg L⁻¹ stock solution of acidic synthetic copper (II) was arranged by dissolving 2.0 g of copper citrate (salts (Cu(NO₃)₂.3H₂O)) and nickel (II) nitrate (salts (Ni(NO₃)₂)) in 1 L of 2 M HCl, with a pH of approximately 2. The calibration standards and test samples were analyzed using an atomic adsorption spectroscope (AA-7000, SHIMADZU).

For the kinetic experiments, an initial concentration of 150 mg L⁻¹ was achieved by diluting the acidic synthetic copper (II) and nickel (II) stock solution. In each Erlenmeyer flask sample, 100 mL of 150 mg L⁻¹ acidic synthetic copper (II) and nickel (II) solution was combined with 0.1 g of each chelating resin Dowex-M4195, and the mixture was stirred at 120 rpm in a water bath at 25 °C. Aliquots were withdrawn at intervals of 5, 10, 15, 30, 60, 120, 180, 240 and 270 min, and subsequently filtrated on Whatman filter 0.45 GF/C for analysis via atomic adsorption spectroscopy.

In terms of the equilibrium adsorption, the isotherms were conducted with 0.1 g of each chelating resin Dowex-M4195 in closed paraffin Erlenmeyer flask samples, exposing them to varying acidic synthetic copper (II) and nickel (II) concentrations ranging from 0 to 250 mg L⁻¹. The samples were agitated in a shaking water bath at 120 rpm for 24 h to reach equilibrium. After filtration, the aliquot part was introduced into the atomic adsorption spectroscope for analysis of the removal of copper (II) and nickel (II).

The equilibrium adsorption and adsorption at time t capacity for each sample were calculated as follows:

$${\mathrm{q}}_{\mathrm{e}}={\displaystyle \frac{({\mathrm{C}}_0-{ \mathrm{C}}_{\mathrm{e}})}{\mathrm{m}}}\mathrm{V}$$

(1)

$${\mathrm{q}}_{\mathrm{t}}={\displaystyle \frac{({\mathrm{C}}_0-{\mathrm{C}}_{\mathrm{e}})}{\mathrm{m}}}\mathrm{V}$$

(2)

where C₀ (mg L⁻¹) is the initial concentration of acidic synthetic copper (II) and nickel (II), C_e (mg L⁻¹) is the concentration of acidic synthetic copper (II) and nickel (II) at equilibrium, V (L) is the volume of the solution used, and m (g) is the mass of chelating resin used.

2.3. Statical Analysis and Its Appropriate Selection

A prevailing trend in the literature underscores the common observation that augmenting the number of parameters within a model typically results in improved correlations. Nevertheless, this research has elected to employ Akaike’s information criterion (AIC) [27,28] for evaluating the effectiveness of the adsorption isotherm and kinetic models, following the guidelines proposed by Akaike in 1998. This approach was chosen despite the potential influence of the number of parameters, which we have deliberately decided to disregard. According to this methodology, the model with the lowest AIC value is considered the most appropriate. AIC provides a measure of the relative quality of statistical models for a given dataset. It is based on the concept of information entropy and offers a means to balance the complexity of the model against its goodness of fit. Specifically, AIC is calculated using the formula:

$$\mathrm{AIC}=\mathrm{N}\times \ln \left({\displaystyle \frac{\mathrm{SSE}}{\mathrm{N}}}\right)+2\mathrm{K}$$

(3)

In our context, to conduct the AIC’s assessment effectively, it is crucial to determine the error sum of squares (SSE). The plea is a fundamental component in this process as it quantifies the deviation of the experimental data from the values predicted by the model. This is achieved by summing the squares of the differences between the observed (experimental) values and the corresponding predicted values, as mathematically represented by the equation:

$$\mathrm{SSE}={{\displaystyle \sum \left({\mathrm{q}}_{\mathrm{e}}-{\mathrm{q}}_{\mathrm{e},\mathrm{pre}}\right)}}^2$$

(4)

Table 3 provides a detailed enumeration of the SSE values for the models under consideration. By integrating these SSE values into the AIC formula, we can objectively compare the relative performance of different adsorption isotherm and kinetic models, thereby identifying the model that best describes the observed data while accounting for the model complexity.

In this analysis, ‘K’ denotes the number of parameters in the model, and ‘N’ represents the number of data points in the dataset. A thorough examination of Table 3 reveals that all the kinetic and isotherm models’ methods consistently yields lower AIC values. This observation underscores the superior efficacy of the FL-PFO method in fitting the data under the given conditions. However, it is important to note that when the dataset comprises fewer than 40 data points, the standard AIC may not provide an accurate assessment due to the finite sample size. In such cases, it is recommended to utilize the corrected Akaike’s information criterion (AIC_corrected), which adjusts for the small sample size and provides a more reliable evaluation of model performance. The AIC_corrected is calculated using the following formula:

$$\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}}=\mathrm{AIC}+{\displaystyle \frac{2\mathrm{K}(\mathrm{K}+1)}{\mathrm{N}-\mathrm{K}-1}}$$

(5)

2.4. Cost and Benefit Analysis

A cost and benefit analysis is a methodical approach employed by water or wastewater treatment plants to evaluate decision options in engineering process design [29,30,31,32]. A cost–benefit analysis was conducted to assess the removal of copper using chelating resin Dowex-M4195 as an adsorbent via the ion-exchanger method [33]. The standard Mishan cost and benefit analysis method [34] was employed for this evaluation, as show the following formula:

$${\displaystyle \frac{\mathrm{B}}{\mathrm{C}}}={\displaystyle \frac{{\displaystyle \sum _{\mathrm{t}=0}^{\mathrm{n}}{\mathrm{B}}_{\mathrm{t}}}}{{\displaystyle \sum _{\mathrm{t}}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{t}}}}}>1$$

(6)

The net social benefit (NSB) [35,36] values representative of the removal of heavy metals in acidic solutions signifies the augmented consumption value resulting from the research project on water and wastewater treatment. This assessment encompasses diverse costs and benefits of decision-making that would arise from the implementation. The NSB (the desired payment (dp) minus indemnified cost (ic)) was determined using the following formula:

$$\mathrm{N}\mathrm{S}\mathrm{B}=\mathrm{d}\mathrm{p}-\mathrm{i}\mathrm{c}$$

(7)

The assessment of both heavy metal removal methods involved an examination of the intangible but significant indirect advantages for human health and the environment. The primary focus of the evaluation was on the responsibilities of water and wastewater treatment plants, and this was conveyed through the presentation of quantitative data delivered verbally.

3. Results and Discussion

3.1. Kinetic Adsorption

In this investigation, the examination of the adsorption kinetics pertaining to the removal of heavy metals of Cu²⁺ and Ni²⁺ solution onto polymeric chelating resin is of paramount importance for elucidating the underlying adsorption mechanisms and acquiring crucial design insights for practical applications. Various kinetic models, namely the pseudo-first-order (PFO), pseudo-second-order (PSO), general (rational) order, Elovich, diffusion chemisorption, and Avrami fractional order models, were employed in the analysis. To mitigate issues associated with data linearization, exclusively nonlinear regression methods were utilized throughout this study.

The determination of the most suitable kinetic model was conducted through a meticulous comparison of the individual and mean numerical values of three critical statistical measures: the adjusted coefficient of determination (R²_adj), the reduced chi-square (Reduced χ²), and the error sum of squares (SSE). These metrics collectively offer a comprehensive assessment of the model’s performance and goodness of fit.

Table 1. Kinetic modeling results of Cu²⁺ and Ni²⁺ adsorption on weakly base polymeric chelating resin Dowex M-4195.

Model	Parameter	Cu2+	Ni2+
PFO [37]
$q_t=q_e\left(1-exp(-k_1\cdot t)\right)$	qe (mg g−1)	44.5638	35.7999
	k1 (min−1)	0.1403	0.0557
	R2adj	0.9718	0.9867
	Reduced χ2	4.1769	1.6256
	SSE	54.3001	21.1330
PSO [38]
$q_t={\displaystyle \frac{k_2\cdot q_e^2\cdot t}{1+q_e\cdot k_2\cdot t}}$	qe (mg g−1)	47.1159	39.3801
	k2 (g mg−1 min−1)	0.0053	0.0020
	R2adj	0.9989	0.9859
	Reduced χ2	0.1577	1.7245
	SSE	2.0503	22.4187
General (rational) order [39]
$q_t=q_e-{\displaystyle \frac{q_e}{{\left[t\cdot k_r\cdot q_e^{n-1}\cdot \left(n-1\right)+1\right]}^{1/(n-1)}}}$	qe (mg g−1)	47.4304	36.6441
	kr [h−1 (g mg−1)n−1]	0.0039	0.0188
	n	2.0888	1.3477
	R2adj	0.9989	0.9902
	Reduced χ2	0.1606	1.1994
	SSE	1.9267	14.3923
Elovich [40,41]
$q_t={\displaystyle \frac{1}{{\beta }_e}}\cdot ln(1+{\alpha }_e\cdot {\beta }_e\cdot t)$	αe (mg g−1 min−1)	859.5738	11.6502
	βe (g mg−1)	0.2202	0.1528
	R2adj	0.9700	0.9409
	Reduced χ2	4.4300	7.2262
	SSE	57.5904	93.9404
Diffusion chemisorption [42]
$q_t={\displaystyle \frac{q_e\cdot k_{DC}\cdot t^{1/2}}{k_{DC}\cdot t^{1/2}+q_e}}$	qe (mg g−1)	53.2206	52.0400
	kDC (mg g−1 min−0.5)	29.5835	9.9001
	R2adj	0.9881	0.9492
	Reduced χ2	1.7675	6.2157
	SSE	22.9777	80.8049
Avrami fractional order [43]
$q_t=q_e\cdot \left[1-\exp {(-(k_{av}\cdot t))}^{n_{av}}\right]$	qe (mg g−1)	44.5634	35.9428
	kav (min−1)	0.1112	0.5432
	nav	1.2621	0.0986
	R2adj	0.9694	0.9998
	Reduced χ2	4.5250	0.7914
	SSE	54.3001	9.4965

Note: q_ex = 36.03 mg g⁻¹(Ni²⁺) and 46.02 mg g⁻¹ (Cu²⁺).

encapsulates the outcomes derived at three distinct temperatures for the six selected models, providing a comprehensive assessment of their efficacy in characterizing the adsorption kinetics of a heavy metal solution onto on weakly base polymeric chelating resin Dowex M-4195.

The analysis of various kinetic models for the adsorption of Cu²⁺ and Ni²⁺ onto weakly base polymeric chelating resin Dowex M-4195 provides significant insights into the adsorption mechanisms and the effectiveness of these models. The pseudo-first-order (PFO) model, with an adjusted R² (R²_adj) of 0.9718 for Cu²⁺ and 0.9867 for Ni²⁺, shows reasonable accuracy in describing the adsorption kinetics. However, its relatively high reduced chi-square (reduced χ²) values of 4.1769 for Cu²⁺ and 1.6256 for Ni²⁺, along with its error sum of squares (SSE) values of 54.3001 and 21.1330, respectively, indicate that there are more accurate models available.

The pseudo-second-order (PSO) model performs exceptionally well, with R²_adj values of 0.9989 for Cu²⁺ and 0.9859 for Ni²⁺. This model’s low reduced χ² values of 0.1577 for Cu²⁺ and 1.7245 for Ni²⁺, and SSE values of 2.0503 and 22.4187, respectively, suggest that it provides a very close fit to the experimental data, particularly for Cu²⁺. The high accuracy of the PSO model indicates that the adsorption process is likely governed by chemisorption, involving electron sharing or exchange between adsorbent and adsorbate.

The general (rational) order model also demonstrates high accuracy, with R²_adj values of 0.9989 for Cu²⁺ and 0.9902 for Ni²⁺. Its reduced χ² values are 0.1606 for Cu²⁺ and 1.1994 for Ni²⁺, and its SSE values are 1.9267 and 14.3923, respectively. These results suggest that this model is also highly suitable for describing the adsorption kinetics, with a slight edge over the PSO model for Ni²⁺ due to its lower reduced χ² and SSE values.

The Elovich model, characterized by high initial adsorption rates (α_e) of 859.5738 mg g⁻¹ min⁻¹ for Cu²⁺ and 11.6502 mg g⁻¹ min⁻¹ for Ni²⁺, shows moderate performance with R²_adj values of 0.9700 for Cu²⁺ and 0.9409 for Ni²⁺. However, its high reduced χ² values of 4.4300 for Cu²⁺ and 7.2262 for Ni²⁺, and SSE values of 57.5904 and 93.9404, respectively, indicate that it is less accurate than the other models considered.

The diffusion chemisorption model shows good performance, with R²_adj values of 0.9881 for Cu²⁺ and 0.9492 for Ni²⁺. Its reduced χ² values are 1.7675 for Cu²⁺ and 6.2157 for Ni²⁺, and its SSE values are 22.9777 and 80.8049, respectively. While this model is better than the Elovich model, it still falls short compared to the PSO and general order models.

Finally, the Avrami fractional order model exhibits excellent performance for both Cu²⁺ and Ni²⁺, with R²_adj values of 0.9694 for Cu²⁺ and an impressive 0.9998 for Ni²⁺. The reduced χ² values are 4.5250 for Cu²⁺ and 0.7914 for Ni²⁺, and the SSE values are 54.3001 and 9.4965, respectively. This model’s superior fit for Ni²⁺ suggests that it effectively captures the complexities of the adsorption process, potentially involving multiple adsorption mechanisms and site heterogeneity.

In conclusion, while the PSO and general order models are highly accurate for Cu²⁺ adsorption, the Avrami fractional order model is particularly effective for Ni²⁺, providing the best fit among the models analyzed. These findings highlight the importance of selecting appropriate kinetic models for accurately describing the adsorption processes of different heavy metals, which is crucial for optimizing the design and operation of adsorption systems in water treatment applications.

Figure 2 provides a visual representation of how the kinetic data align with the six kinetic models discussed, ultimately demonstrating the superior performance of the PSO and general order models. As a best practice, it is essential to conduct a thorough examination of predicted versus measured data, as well as residual plots (although not included here), to identify any discernible trends that may indicate the unsuitability of the chosen models. Such analyses contribute to the robustness of the conclusions drawn from the study.

3.2. Adsorption Isotherm

Sorption equilibrium is achieved when the interaction between the adsorbent and the adsorbate persists over an extended period, allowing a stable mass balance to be established between the concentration of the adsorbate in the bulk solution and the amount of adsorbate accumulated on the surface of the adsorbent. This equilibrium state indicates that the rates of adsorption and desorption are equal, leading to a steady-state condition where no further net adsorption occurs. Therefore, the mechanism of adsorption needs to be explored. Numerous empirical and semi-empirical isotherm models exist for predicting equilibrium conditions. In this research, an exploration of several models was conducted to quantify the adsorption characteristics of the system, as shown in

Table 2. Isotherm modeling results of Cu²⁺ and Ni²⁺ adsorption on weakly base polymeric chelating resin Dowex M-4195.

Model	Parameter	Cu2+	Ni2+
Freundlich
$q_e=k_f\cdot C_e{ }^{1/n_f}$	kf (mg g−1)(L g−1)n	17.0593	10.8396
	nf	3.7965	3.5675
	1/nf	0.2634	0.2803
	R2adj	0.8582	0.9045
	Reduced χ2	46.5335	17.1315
	SSE	418.8013	154.1836
Langmuir
$q_e=q_m\cdot k_l{\displaystyle \frac{C_e}{1+k_l\cdot C_e}}$ $R_l={\displaystyle \frac{1}{1+k_l\cdot C_o}}$	qm (mg g−1)	47.9184	36.4825
	kl (L MB−1 mg−1)	0.4556	0.2233
	Rl	0.0215	0.0429
	R2adj	0.9742	0.9925
	Reduced χ2	8.4493	1.3477
	SSE	76.0442	12.1293
Temkin [44]
$q_e=b_t\cdot \left|\ln \left(k_t\cdot C_e\right)\right|$	bt (J mol−1)	0.7815	6.4899
	kt (L mol−1)	2.2372	3.3449
	R2adj	0.9375	0.9648
	Reduced χ2	20.5152	6.3223
	SSE	184.6371	56.9039
Harkin–Jura [45]
$q_e={\left({\displaystyle \frac{a_H}{b_H-log{C}_e}}\right)}^{1/2}$	aH	110.4031	79.5571
	bH	2.8917	2.9862
	R2adj	0.7823	0.8332
	Reduced χ2	71.4453	29.9164
	SSE	643.0076	269.2474
Jovanovic [46]
$q_e=q_m\left(1-ex{p}^{(k_j\cdot C_e)}\right)$	qm (mg g−1)	44.0896	33.1401
	kj	0.3787	0.1774
	R2adj	0.9775	0.9929
	Reduced χ2	7.3995	1.2638
	SSE	66.5658	11.3739
Toth [47]
$q_e=q_m{\displaystyle \frac{k_{th}\cdot C_e}{{\left(1+{(k_{th}\cdot C_e)}^{n_{th}}\right)}^{1/n_{th}}}}$	qm (mg g−1)	44.7519	34.4145
	kth	0.3379	0.1742
	nth	1.7365	1.42925
	R2	0.9802	0.9953
	Reduced χ2	6.5108	0.8381
	SSE	52.0867	6.7047

Note: q_ex = 32.86 mg g⁻¹ (Ni²⁺) and 44.37 mg g⁻¹ (Cu²⁺).

, offering insights into their respective predictive capabilities.

In the pursuit of understanding the adsorption isotherm of solute uptake onto weakly base polymeric chelating resin Dowex M-4195, the process design hinges on the meticulous determination of the isotherm models. Traditionally, linear regression with the least squares method has been the go-to approach, despite its limitations. Linearizing nonlinear equations through this method not only alters the error structure but also contravenes the fundamental assumptions of linear regression. Hence, this research opted for nonlinear regression employing the sum of squared residuals as the error function. This enabled a more accurate assessment of various models to characterize the adsorption process isotherms and shed light on the underlying controlling mechanisms. Table 2 encapsulates the key parameters of six adsorption isotherm models that integrate essential statistical metrics obtained from nonlinear regression analyses. This detailed presentation facilitates a thorough comparative evaluation of the models, thereby guiding the selection of the most appropriate model for describing the adsorption phenomena under study.

The analysis of various isotherm models for the adsorption of Cu²⁺ and Ni²⁺ onto weakly base polymeric chelating resin Dowex M-4195 provides significant insights into the adsorption mechanisms and the effectiveness of these models.

The Freundlich isotherm model, characterized by its capacity to describe adsorption on heterogeneous surfaces, shows moderate accuracy with adjusted R² (R²_adj) values of 0.8582 for Cu²⁺ and 0.9045 for Ni²⁺. However, the high reduced chi-square (reduced χ²) values of 46.5335 for Cu²⁺ and 17.1315 for Ni²⁺, along with high root mean square error (SSE) values of 418.8013 and 154.1836, respectively, indicate significant deviations from the experimental data, making it less suitable for precise modeling in this context.

The Langmuir isotherm model, which assumes monolayer adsorption on a homogeneous surface, performs exceptionally well. It has high R²_adj values of 0.9742 for Cu²⁺ and 0.9925 for Ni²⁺, indicating an excellent fit to the experimental data. The low reduced χ² values of 8.4493 for Cu²⁺ and 1.3477 for Ni²⁺, along with relatively low SSE values of 76.0442 and 12.1293, further support the model’s accuracy. The Langmuir model’s maximum adsorption capacity (q_m) values of 47.9184 mg g⁻¹ for Cu²⁺ and 36.4825 mg g⁻¹ for Ni²⁺ reflect its effectiveness in describing the adsorption process.

The Temkin isotherm model, which considers adsorbent–adsorbate interactions, shows good performance, with R²_adj values of 0.9375 for Cu²⁺ and 0.9648 for Ni²⁺. However, its reduced χ² values of 20.5152 for Cu²⁺ and 6.3223 for Ni²⁺, and SSE values of 184.6371 and 56.9039, respectively, indicate that it is less accurate than the Langmuir model but still a viable option for modeling adsorption.

The Harkin–Jura isotherm model, which is often used for multi-layer adsorption on heterogeneous surfaces, shows the least accuracy among the models analyzed. It has low R²_adj values of 0.7823 for Cu²⁺ and 0.8332 for Ni²⁺, and extremely high reduced χ² values of 71.4453 and 29.9164, and SSE values of 643.0076 and 269.2474, respectively. These results indicate significant deviations and a poor fit to the experimental data, making this model unsuitable for describing the adsorption processes of Cu²⁺ and Ni²⁺ on Dowex M-4195.

The Jovanovic isotherm model, which assumes a more realistic scenario of localized adsorption without interactions between adsorbed molecules, performs very well, with R²_adj values of 0.9775 for Cu²⁺ and 0.9929 for Ni²⁺. The low reduced χ² values of 7.3995 for Cu²⁺ and 1.2638 for Ni²⁺, along with SSE values of 66.5658 and 11.3739, indicate a close fit to the experimental data, making it a strong contender alongside the Langmuir model.

The Toth isotherm model, which accounts for both heterogeneity and the possibility of multi-layer adsorption, shows excellent performance, with R² values of 0.9802 for Cu²⁺ and 0.9953 for Ni²⁺. It has the lowest reduced χ² values of 6.5108 for Cu²⁺ and 0.8381 for Ni²⁺, and SSE values of 52.0867 and 6.7047, respectively, suggesting that it provides the best fit among all the models analyzed. The Toth model’s maximum adsorption capacity (q_m) values of 44.7519 mg g⁻¹ for Cu²⁺ and 34.4145 mg g⁻¹ for Ni²⁺ further validate its suitability.

In conclusion, while the Langmuir and Jovanovic models provide highly accurate descriptions of the adsorption processes for both Cu²⁺ and Ni²⁺, the Toth model stands out as the most effective, particularly for Ni²⁺, due to its ability to account for surface heterogeneity and multi-layer adsorption. The Freundlich and Harkin–Jura models are less accurate and less suitable for this system. These findings highlight the importance of selecting appropriate isotherm models to accurately describe adsorption mechanisms, which is crucial for optimizing the design and operation of adsorption systems in water treatment applications.

In Figure 3, a comprehensive display of the fits to various two-parameter and three-parameter adsorption isotherms is presented, meticulously produced to ensure optimal visual clarity. This curated selection allows for a detailed examination of the models’ performance in capturing the experimental data, providing insights into the suitability of each model for describing the adsorption process. By showcasing a range of models, from simpler two-parameter ones to more complex three-parameter ones, the figure offers a nuanced understanding of the underlying mechanisms governing adsorption behavior. This visual representation serves as a valuable tool for researchers and practitioners alike in assessing the appropriateness of different models for their specific applications, ultimately enhancing the understanding and optimization of adsorption processes.

It could be inferred that ion exchange was likely the primary mechanism for Cu²⁺ and Ni²⁺ adsorption, as illustrated in Figure 4a,b. This process can be represented by the following reaction: (RH_n + Cu²⁺ ⇄ R-Cu²⁺ + nH⁺). This equation indicates that the resin (RH_n) interacts with the Cu²⁺ or Ni²⁺ ion in the solution, resulting in the formation of a resin–metal complex (R-Cu²⁺ or R-Ni²⁺) and the release of hydrogen ions (H⁺). However, the adsorption mechanism of Dowex M-4195 for the removal of Cu²⁺ and Ni²⁺ from water has been extensively detailed in a previous study [19].

This mechanism is supported by the observed adsorption behavior, where the replacement of hydrogen ions on the resin with copper ions from the solution plays a pivotal role. The efficiency of this ion-exchange process is critical for the effective removal of Cu²⁺ and Ni²⁺ from aqueous environments, highlighting the resin’s capability to selectively adsorb metal ions through this exchange mechanism.

In comparison, the proposed method using Dowex M-4195 represents a significant improvement in heavy metal removal, particularly in acidic environments. Traditional adsorbents like activated carbon and chitosan often underperform in low pH conditions due to the reduced efficiency and selectivity [1,48]. In contrast, Dowex M-4195 maintains high adsorption capacities for metals such as copper and nickel, even in challenging conditions, due to its bispicolamine functional group, which enhances its stability and selectivity [12].

Furthermore, Dowex M-4195 is advantageous because it can be regenerated and reused without significant efficiency loss, unlike activated carbon, which degrades with harsh regeneration processes [11]. Its alignment with circular economy principles supports sustainable resource management, making it a superior alternative for modern applications [5].

3.3. Statical Analysis and Its Appropriated Results

The analysis of Akaike’s information criterion (

Table 3. Akaike’s information criterion for six kinetic adsorption and isotherm models of Cu²⁺ adsorption on weakly base polymeric chelating resin Dowex M-4195.

Models	N	K	SSE	AIC	AICcorrected
Kinetic adsorption models
PFO	15	2	54.30	23.30	24.30
PSO	15	2	2.05	−25.85	−24.85
General (rational) order	15	3	1.93	−24.76	−22.58
Elovich	15	2	57.59	24.18	25.18
Avrami fractional order	15	3	54.30	25.30	27.48
Diffusion chemisorption	15	2	22.98	10.40	11.40
Adsorption isotherm models
Freundlich	11	2	418.80	44.03	45.53
Langmuir	11	2	76.04	25.27	26.77
Temkin	11	2	184.64	35.03	36.53
Harkin–Jara	11	2	643.01	48.75	50.25
Jovanovic	11	2	66.60	23.81	25.31
Toth	11	3	52.09	23.11	26.53

) for the kinetic adsorption and isotherm models of Cu²⁺ adsorption on Dowex M-4195 resin provides significant insights into the model performance. For the kinetic adsorption models, the PSO model emerges as the most suitable, with the lowest AIC (−25.85) and AIC_corrected (−24.85), indicating its high accuracy and better fit compared to the other models. On the contrary, the Elovich model is the least suitable, having the highest AIC (24.18) and AIC_corrected (25.18), suggesting poor performance in describing the adsorption kinetics of Cu²⁺.

In terms of the adsorption isotherm models, the Toth model stands out as the best fit, with the lowest AIC (23.11), although its AIC_corrected (26.53) is slightly higher, indicating a slight correction for a small sample size. The Jovanovic model also shows competitive performance with a slightly higher AIC (23.81) but lower AIC_corrected (25.31) than the Toth model, making it another viable option. Conversely, the Harkin–Jura model exhibits the highest AIC (48.75) and AIC_corrected (50.25), making it the least accurate isotherm model for Cu²⁺ adsorption on the resin.

In conclusion, the PSO and Toth models are recommended for accurately predicting Cu²⁺ adsorption kinetics and isotherms, respectively, which confirms and consists of the results in Section 3.1 and Section 3.2. These models’ lower AIC values suggest a better fit and reliability, which are crucial for optimizing adsorption processes in various applications. Conversely, the Elovich and Harkin–Jura models are less suitable and may require refinement or replacement in future studies. This comparative analysis underscores the importance of selecting appropriate models to enhance the efficiency and accuracy of adsorption systems in environmental and industrial processes.

Figure 5 illustrates the corrected Akaike’s information criterion (AIC_corrected) values for the Cu²⁺ adsorption isotherm and kinetic adsorption models, providing a rigorous evaluation of the model performance. For the adsorption models, the PSO kinetic and Toth isotherm models are identified as the most suitable models, as evidenced by the lowest AIC_corrected values. This finding confirms the superior fit of the PSO and Toth models in describing the kinetic and isotherm adsorption behavior of Cu²⁺. This comprehensive analysis underscores the importance of selecting an appropriate model based on the corrected AIC values, ensuring the reliability and validity of the conclusions drawn from the adsorption study.

The analysis of Akaike’s information criterion, as shown in

Table 4. Akaike’s information criterion for six kinetic adsorption and isotherm models of Ni²⁺ adsorption on weakly base polymeric chelating resin Dowex M-4195.

Models	N	K	SSE	AIC	AICcorrected
Kinetic adsorption models
PFO	15	2	21.13	9.14	10.14
PSO	15	2	22.42	10.02	11.03
General (rational) order	15	3	14.39	5.38	7.56
Elovich	15	2	93.94	31.52	32.52
Avrami fractional order	15	3	1.57761 × 10−10	−373.17	−370.99
Diffusion chemisorption	15	2	80.81	29.26	30.26
Adsorption isotherm models
Freundlich	11	2	154.18	33.04	34.54
Langmuir	11	2	12.135	5.08	6.58
Temkin	11	2	56.90	22.08	23.58
Harkin–Jara	11	2	269.25	39.18	40.68
Jovanovic	11	2	11.37	4.36	5.86
Toth	11	3	6.70	0.55	3.97

, for the kinetic adsorption and isotherm models of Ni²⁺ adsorption on Dowex M-4195 resin reveals that the Avrami fractional order model is the most suitable kinetic model, with the lowest AIC (−373.17) and AIC_corrected (−370.99), indicating its superior accuracy and fit. Conversely, the Elovich model is the least suitable for describing the adsorption kinetics of Ni²⁺, as indicated by its highest AIC (31.52) and AIC_corrected (32.52). Among the adsorption isotherm models, the Toth model demonstrates the best fit with the lowest AIC (0.55) and AIC_corrected (3.97), while the Harkin–Jura model performs the worst, having the highest AIC (39.18) and AIC_corrected (40.68). These findings underscore the importance of selecting the appropriate models for accurately predicting and optimizing Ni²⁺ adsorption processes. The Avrami fractional order and Toth models are recommended for their superior performance, while the Elovich and Harkin–Jura models may require refinement or replacement in future studies. This comparative analysis is crucial for enhancing the efficiency and reliability of adsorption systems in various environmental and industrial applications.

The comprehensive analysis provided in Figure 6 underscores the importance of using AIC_corrected values in model selection, enhancing the reliability and validity of the adsorption study’s conclusions. This detailed visual representation serves as a valuable tool for researchers in evaluating and comparing the predictive capabilities of various adsorption isotherm and kinetic models.

Based on the provided bar chart comparing the average absolute relative deviation (AARD%) in Figure 7 for Cu²⁺ and Ni²⁺ adsorption using various kinetic models, we can draw several key conclusions. The pseudo-second-order (PSO) model shows the lowest AARD% for Cu²⁺ (0.77%), indicating it is highly accurate for modeling Cu²⁺ adsorption. However, for Ni²⁺, the PSO model has a higher AARD% (3.77%), indicating a lesser fit for Ni²⁺ adsorption. Similarly, the general order model also demonstrates low AARD% for Cu²⁺ (0.76%), making it another accurate model, whereas for Ni²⁺, it performs moderately, with an AARD% of 3.38%.

The Avrami fractional order model stands out for Ni²⁺, exhibiting an extremely low AARD% (essentially zero), making it the most accurate model for Ni²⁺ adsorption. Conversely, this model shows moderate performance for Cu²⁺ with an AARD% of 4.03%. The Elovich model, on the other hand, performs poorly for both ions, with the highest AARD% for Cu²⁺ (8.09%) and a high value for Ni²⁺ (4.14%). The diffusion chemisorption model indicates moderate accuracy for Cu²⁺ adsorption, with an AARD% of 2.58%, but poor performance for Ni²⁺ adsorption, with a high AARD% of 7.98%. Lastly, the pseudo-first-order (PFO) model shows similar moderate accuracy for both Cu²⁺ and Ni²⁺, with AARD% values of 4.03% and 3.98%, respectively.

Overall, the PSO and general order models are recommended for studies focused on Cu²⁺ adsorption, while the Avrami fractional order model is most suitable for Ni²⁺ adsorption that is related to the results of the error function, as illustrated in Section 3.1 and Section 3.2 and the AIC_corrected results. Further research should be conducted to validate these findings with additional data and to investigate the reasons behind the high AARD% for the Elovich model, potentially exploring improvements or alternative models. These insights can guide the design of adsorption systems and the prediction of Cu²⁺ and Ni²⁺ behavior in various environmental and industrial processes.

The bar chart comparing the AARD% for Cu²⁺ and Ni²⁺ adsorption using various isotherm models, as shown in Figure 8, reveals that the Toth model is the most accurate for Cu²⁺ adsorption, with the lowest AARD% at 5.55%, while the Langmuir model is the best for Ni²⁺ adsorption, with an AARD% of 14.87%. Conversely, the Harkin–Jura model exhibits the highest AARD% for both Cu²⁺ (36.93%) and Ni²⁺ (54.1%), indicating it is the least accurate for both ions. Other models, such as the Freundlich, Temkin, and Jovanovic models, show varying degrees of accuracy, with moderate performance for both ions. Therefore, for the accurate prediction of Cu²⁺ and Ni²⁺ adsorption, the Toth and Langmuir models are recommended, respectively. Further research should aim to validate these findings with additional data and explore improvements for the less accurate models like the Harkin–Jura model. These insights are essential for designing effective adsorption systems and predicting the behavior of these ions in various environmental and industrial processes.

3.4. Social Cost–Benefit Analysis

Table 5. Quantities of chemicals used in studying isotherm and kinetic adsorption for measuring the removal and cost per unit in the chemical removal process of Cu²⁺ and Ni²⁺.

Chemical Name Used	Cu2+		Ni2+
	Amont of Chemical Used (g or L)	Current Cost (2024, THB or USD)	Amont of Chemical Used (g or L)	Current Cost (2024, THD or USD)
Sodium hydroxide (5 mol L−1, >97%, Panreac)	Average 0.001	THB 0.98, USD 0.027	Average 0.001	THB 0.98, USD 0.027
Acetic acid (99.99%, AR grade, Ajex Finechem)	Average 0.002	THB 0.73, USD 0.020	Average 0.002	THB 0.73, USD 0.020
Dowex M4195 (Sigma-Aldrich)	Fixed 0.1	THB 16.60, USD 0.47	Fixed 0.1	THB 16.60, USD 0.47
Total cost (THB)	-	18.31	-	18.31
Total cost (USD)	-	0.517	-	0.517
Model and cost analysis	Maximum adsorption (Cu2+ and Ni2+ removal)
Kinetic adsorption	46.02 mg g−1 (B, Benefit = 46.02)		36.03 mg g−1 (B, Benefit = 36.03)
Adsorption isotherm	44.37 mg g−1 (B, Benefit = 44.37)		32.86 mg g−1 (B, Benefit = 32.86)
B/C Kinetic	46.02/18.31 = 2.513 (Baht)		36.03/18.31 = 1.967 (Baht)
B/C Isotherm	44.37/18.31 = 2.423 (Baht)		32.86/18.31 = 1.795 (Baht)
$\sum \mathrm{N}\mathrm{S}\mathrm{B}$ (net socail benefit) kinetic and isotherm	18.31 − (18.31/5) = 14.648 (baht)

presents the cost–benefit analysis for the adsorption of Cu²⁺ and Ni²⁺ using weakly base polymeric chelating resin Dowex M-4195, which demonstrates a clear economic advantage in the kinetic adsorption process compared to the isotherm adsorption process. The study examined the quantities and costs of the chemicals used, including sodium hydroxide, sulfuric acid, and Dowex M-4195, with a total cost of THB 18.31 (USD 0.517) for both metals. The maximum adsorption capacities were 46.02 mg g⁻¹ for Cu²⁺ and 36.03 mg g⁻¹ for Ni²⁺ in the kinetic adsorption process, and 44.37 mg g⁻¹ for Cu²⁺ and 32.86 mg g⁻¹ for Ni²⁺ in the isotherm adsorption process. The benefit–cost (B/C) ratios suggest that the kinetic adsorption process is more costly to implement, with B/C ratios of 2.513 for Cu²⁺ and 1.967 for Ni²⁺, compared to the isotherm adsorption process, which has B/C ratios of 2.423 for Cu²⁺ and 1.795 for Ni²⁺. The net social benefit (NSB) for both processes is calculated to be THB 14.648.

In conclusion, the kinetic adsorption models, particularly the PSO and general order models, offer a superior fit and higher adsorption efficiency for the removal of Cu²⁺ and Ni²⁺. While the isotherm models, especially the Langmuir and Toth models, are slightly less efficient, they still provide significant economic benefits. These findings underscore the importance of selecting appropriate adsorption models to enhance both the efficiency and economic viability in heavy metal removal processes. By integrating these insights into a circular economy framework, the sustainability of the process can be further improved through the reuse and recycling of adsorbents and recovered metals, leading to more effective water treatment solutions and reduced environmental and health risks.

3.5. The Significance of the Research

The innovation and significance of this research introduce a groundbreaking approach to heavy metal removal from acidic solutions through the innovative use of the weakly base polymeric chelating resin Dowex M-4195. By integrating advanced adsorption models with comprehensive statistical analyses, this study offers a deeper understanding of the adsorption kinetics and isotherms specific to copper and nickel ions, setting a new standard for efficient and sustainable water treatment technologies.

The study’s significance lies in its potential to revolutionize heavy metal remediation practices, particularly in challenging acidic environments. Dowex M-4195’s superior performance provides a solid foundation for future research and practical applications, especially in industries dealing with metal contamination. The inclusion of circular economy principles further enhances the study’s impact, promoting sustainable resource management and environmental protection.

The evaluation of various adsorption isotherm and kinetic models using AIC reveals that the optimal model varies depending on the specific heavy metal ion and adsorbent, highlighting the complexity of adsorption processes. This suggests that a one-size-fits-all approach is inadequate for accurately modeling adsorption across different systems. Notably, this study identifies models like the Avrami fractional order model, which, despite being less common in experimental research, may offer superior fits for certain scenarios.

The practical significance of this evaluation process is its ability to identify the most accurate models for specific adsorption systems, thereby improving the precision of experimental designs and industrial applications in water treatment and pollutant removal. However, conducting exhaustive evaluations for each adsorbent–metal ion combination can be economically and labor-intensive. To address this, this study suggests developing guidelines based on effective models identified in comprehensive evaluations, streamlining the model selection process for future research.

While full-scale evaluations may not always be feasible, applying insights from comprehensive studies to similar systems can reduce the effort required and enhance the efficient use of adsorption models in both research and industry.

4. Conclusions

This study thoroughly investigates the removal of heavy metals, particularly Cu²⁺ and Ni²⁺, from acidic aqueous solutions using the weakly base polymeric chelating resin Dowex M-4195. The key findings reveal that the PSO and general order models are highly accurate in describing the adsorption kinetics of Cu²⁺, with the PSO model showing superior performance. For Ni²⁺ adsorption, the Avrami fractional order model provides the best fit among the kinetic models analyzed. In terms of the isotherm models, the Langmuir and Jovanovic models offer highly accurate descriptions for both Cu²⁺ and Ni²⁺, while the Toth model is most effective for Ni²⁺ due to its ability to account for surface heterogeneity and multi-layer adsorption. The Freundlich and Harkin–Jura models are less suitable for this system.

The cost–benefit analysis indicates that the kinetic adsorption process is more economically beneficial than the isotherm adsorption process, with benefit–cost (B/C) ratios of 2.513 for Cu²⁺ and 1.967 for Ni²⁺ in kinetic adsorption, compared to 2.423 for Cu²⁺ and 1.795 for Ni²⁺ in isotherm adsorption. The total cost of chemicals used for the adsorption process is THB 18.31 (USD 0.517). Integrating the principles of a circular economy enhances the sustainability of the adsorption process, as Dowex M-4195 can be regenerated and reused, and the metals captured by the resin can potentially be recovered and reused in industrial processes.

Selecting the appropriate kinetic and isotherm models is crucial for accurately predicting and optimizing the adsorption processes of heavy metals, which is essential for designing efficient and reliable adsorption systems in environmental and industrial applications. This study’s findings provide valuable insights into the adsorption mechanisms and effectiveness of Dowex M-4195, guiding future research and practical applications in water treatment and pollution control. By combining thorough statistical analysis with cost–benefit considerations and circular economy principles, this study offers a robust framework for improving water treatment processes, promoting environmental sustainability, and ensuring the health and safety of communities.