Valorisation of banana peel waste for copper (II) and chromium (VI) biosorption from aqueous solution: classical and fractal-like kinetics and isotherms model analysis

1 Introduction

The discharge of heavy metals into aquatic environments remains a persistent global challenge driven largely by rapid industrialisation, urban expansion, and inadequate wastewater treatment infrastructure particularly in sub-Saharan Africa (Mkhonza et al., 2026). Among the various toxic metals of concern, copper (Cu), chromium (Cr), lead (Pd), mercury (Hg), cadmium (Cd), arsenic (As), and nickel (Ni) are particularly problematic due to their high mobility, and severe ecological and human health impacts (Mkhonza et al., 2026; Onu et al., 2023). Available literature suggests that, the occurrence and relatively high detection factor of Cu(II) and Cr(VI) in aquatic environment is attributed to their high solubility and mobility (Mkhonza et al., 2026; Sawicka et al., 2021). Cu(II), although an essential micronutrient at trace levels, becomes toxic at elevated concentrations, contributing to gastrointestinal distress, liver damage, and oxidative stress (Gaetke and Chow, 2003). Cr(VI) on the other hand, is a well-established carcinogen associated with mutagenicity, respiratory complications, and organ failure even at low exposure levels (Sawicka et al., 2021). Therefore, the persistence of these metals in water bodies, coupled with their non-biodegradable nature, suggests that there is an urgent need for sustainable, cost-effective, and efficient remediation strategies. Conventional water treatment technologies, viz., chemical precipitation, ion exchange, membrane filtration, and electrochemical methods have demonstrated varying degrees of success in removing heavy metals from aqueous systems (Nemeş et al., 2026; Fei and Hu, 2023). However, these methods often suffer from high operational costs, sludge generation, energy intensity, and limited applicability in low-resource settings, making them less suitable for widespread adoption in developing regions. As a result, research attention has increasingly shifted toward biosorption, a low-cost, environmentally friendly, and highly effective technique that utilises biological materials to sequester metal ions from aqueous environments.

Recently, agricultural-waste-derived biosorbents have emerged as promising alternatives to conventional adsorbents due to their abundance, renewability, and rich surface chemistry (Ngomane et al., 2026; Sibiya-Dlomo et al., 2026). Agricultural residues such as banana peels (BPs), orange peels, coconut husks, rice husks, and sawdust contain lignocellulosic components, cellulose, hemicellulose, lignin, and pectin that provide functional groups capable of binding metal ions (Malik et al., 2017; Ungureanu et al., 2023). Among these materials, BP waste has gained significant attention due to its global availability, low cost, and favourable physicochemical properties (Ngomane et al., 2026). Banana production generates substantial quantities of peel waste, often exceeding 30%–40% of the fruit mass which is typically discarded in landfills, contributing to greenhouse gas emissions and environmental degradation (Alzate Acevedo et al., 2021; Mkhize et al., 2025). Studies on BP suggest that they possess a naturally porous structure and contain abundant functional groups such as hydroxyl, carboxyl, carbonyl, amine moieties, which facilitate adsorbate binding through mechanisms, i.e., ion exchange, complexation, electrostatic attraction, and micro-precipitation (Ngomane et al., 2026). To further enhance biosorption performance, researchers have explored chemical activation, thermal modification and surface functionalisation of BP biomass. Chemical activation using acids, bases, or oxidising agents increases surface area, pore volume, and functional groups availability, thereby improving adsorption capacity (Nadew et al., 2023). Nadew, Keana (Nadew et al., 2023) reported that the activation (thermal + chemical) of BP biomass improves the physicochemical characteristics, i.e., surface area increased from 75 m²/g to 432 m²/g and the thermal stability improved with decomposition shifting from 150 °C (raw) to 300 °C (activated).

It is worth noting that, the biosorption performance of BP-derived biosorbents is influenced by several physicochemical parameters, namely, solution pH, initial metal concentration, biosorbent dosage, contact time, agitation speed, and temperature (Rahman et al., 2026). The solution pH is particularly critical, as it affects both metal speciation and the ionisation state of functional groups on the biosorption surface. In solution, Cr(IV), i.e., Cr⁶⁺ exists predominantly as HCr O 4 −, Cr 2 O 7 2 − or Cr O 4 2 − depending on pH, with maximum adsorption typically occurring under acidic conditions where electrostatic attraction is strongest (Nyairo et al., 2025). On the other hand, Cu(II), i.e., Cu²⁺ adsorption is favoured at moderately acidic pH conditions, where competition with hydrogen ions is reduced and binding sites are more accessible (Macena et al., 2025). To elucidate adsorption mechanisms and optimise process conditions, kinetic and isotherm models are widely employed. Kinetic models such as the pseudo-first order (PFO) and pseudo-second order (PSO) provide insights into the rate-controlling steps and whether adsorption is dominated by physisorption or chemisorption (Hu et al., 2022). On the other hand, isotherms play a central role in describing how much of a substance can be adsorbed at equilibrium and how the adsorbate distributes between the solid and liquid phase (Danat et al., 2026). Recent advances in biosorption research have indicated the importance of non-ideal kinetic models, including fractal-like approaches, to capture the heterogeneous and multi-scale nature of biosorption surfaces (Ngcobo et al., 2025). Traditional kinetic models such as PFO and PSO assume uniform surface energy and constant diffusion pathways, which rarely reflect the true behaviour of natural materials.

On the other hand, fractal kinetics introduce a fractal time exponent that accounts for surface irregularities, pore tortuosity, and anomalous diffusion (Kopelman, 1988; Haerifar and Azizian, 2014). This provides a more realistic representation of adsorption processes occurring on complex biosorbent matrices derived from agricultural waste. BP waste, an abundant agricultural by-product, has demonstrated considerable potential as a low-cost biosorbent for heavy metal removal from aqueous environments. Prior investigations have explored Cu and Cr removal individually using various biosorbent matrices (Seleman et al., 2023; Parlayıcı and Baran, 2025), the simultaneous application of classical and fractal-like kinetic and isotherm frameworks to characterise biosorption onto raw BP remains underexplored. Classical kinetic models including the PFO and PSO equations assume homogeneous surface conditions and time-invariant rate constants. However, biosorbent surfaces are inherently heterogeneous, giving rise to fractal-like adsorption dynamics in which the rate coefficient decays as a power function of time (Haerifar and Azizian, 2012). Building on this framework, Haerifar and Azizian (Haerifar and Azizian, 2014) demonstrated that fractal-like kinetics more accurately capture adsorption onto structurally disordered surfaces. As such, the present study advances this body of knowledge by systematically evaluating both classical and fractal-like kinetics models for Cu(II) and Cr(IV) biosorption from aqueous solution using BP waste, providing a mechanistical description of the sorption process and affirming the suitability of low-cost agricultural waste for sustainable water remediation.

2 Materials and methods

2.1 Materials

Herein, all chemicals used were of analytical grade. Cu²⁺ and Cr⁶⁺ ions solutions were prepared by dissolving a specific mass of copper sulphate pentahydrate (Cu₂SO₄.5H₂O) and sodium chromate tetrahydrate (Na₂CrO₄.4H₂O) salts, respectively, supplied by Sigma-Aldrich. BP waste was collected from a local fruit and vegetable market, samples were washed using tap water to removal any particulate matter, thereafter, BPs were rinsed using deionised water. The washed BPs were then placed on a tray and dried using a convection air dryer at a constant air speed of 1.6 m/s and a temperature of 105 °C for 48 h. The dried BPs were then crushed using a blender and reduced to powder, the produced BP powder was then rinsed using deionised water to remove impurities. Thereafter, the resultant cake was dried in an oven at a constant temperature of 120 °C for 48 h until the sample was free from moisture. It is imperative to note that, the resultant BP cake was systematically dried at 120 °C based on thermogravimetric analysis (TGA) reported by Tahir, Zhao (Tahir et al., 2019), who demonstrated through TGA-FTIR characterisation of BP pyrolysis that appreciable mass loss commences below 163 °C due to moisture evaporation, while carboxylic acid group gasification (CH₃OOH) initiates at approximately 122 °C. By drying at 120 °C marginally below the decomposition temperature, surface moisture is effectively removed without devolatilising the lignocellulosic matrix or degrading oxygen-bearing functional groups (i.e., carboxylic, hydroxyl, and carbonyl moieties) essential for Cu²⁺ and Cr⁶⁺ biosorption, thereby preserving biosorbent physicochemical integrity. The particle size of the final BP-derived biosorbent was in the range of 210 μm and 180 µm. The BP-derived biosorbent was then characterised using the Fourier Transform Infrared (FTIR) spectroscopy, scanning electron microscopy-energy dispersive X-ray spectroscopy (SEM-EDS), X-ray diffraction (XRD) analysis, and Brunauer–Emmett–Teller (BET).

2.2 Batch adsorption studies

Batch adsorption experiments were conducted using a standard jar-test apparatus to evaluate the percentage removal of Cu²⁺ and Cr⁶⁺ ions by BP biosorbent under controlled conditions. A predetermined volume of metal solution, i.e., 500 mL at the desired initial adsorbate concentration.

(5–40 mg/L) was transferred into individual 1 L beakers positioned on the jar-test unit. Thereafter, a known mass of BP corresponding to the required adsorbent dosage (1–6 g/L) was added. The initial pH of each solution was adjusted to the desired value (pH range of 2–7) using 1.0 M HCl and/or NaOH prior to commencement of experimental runs. Agitation was then initiated at a predetermined mixing speed of 150 rpm to ensure uniform contact between the BP biosorbent and metal ions, and the adsorption process was allowed to proceed for 140 min with sampling taking place at specific time intervals of 10 min for kinetics studies and after 140 min for isotherm studies. It is imperative to note that the contact time of 140 min was selected to ensure complete metal ion uptake onto the BP biosorbent based on preliminary runs, which demonstrated that Cu²⁺ and Cr⁶⁺ uptake was rapid within the initial 5–20 min, followed by a progressively slower adsorption phase as active binding sites became increasingly saturated, a behaviour consistent with reported literature (Karnwal, 2024). Furthermore, equilibrium was attained at approximately 120–135 min, Cr⁶⁺ exhibited the longest equilibration period, attributable to its sequential reduction to Cr³⁺ prior to surface sorption (Gupta et al., 2025). At each sampling point, aliquots were withdrawn and filtered to separate the BP biosorbent from the aqueous phase. The clear supernatant was subsequently analysed for residual Cu²⁺ and Cr⁶⁺ concentrations using an atomic absorption spectroscopic for Cu²⁺ and inductive coupled plasma mass spectrometry for Cr⁶⁺. All experiments were performed at room temperature and in triplicates to ensure reproducibility, and the adsorption capacity and percentage removal were computed using Equations 1 and 2, respectively. where q exp is the experimental equilibrium adsorption capacity measured in mg/g, C 0 and C f are the initial and final concentrations of adsorbate in solution measured in mg/L, m is the mass of adsorbent in g, and V is the sample volume measured in L.

q exp = C 0 − C f m V (1)

Removal , % = C 0 − C f C 0 × 100 (2)

2.3 Adsorption empirical models

2.3.1 Batch adsorption kinetic models

Adsorption kinetics studies were studied by fitting adsorption experimental data in nonlinear PFO (Equation 3), PSO (Equation 4), mixed-order-equation, (MOE) (Equation 5), pseudo-nth-order, (PNO) (Equation 6), fractal-like PFO (Equation 7), and fractal-like PSO (Equation 8) as discussed by Hu et al. (2022); Ngcobo et al. (2025). where, “q_t and q_e are the amounts of adsorbate uptake per unit mass of the adsorbent at time t and at equilibrium measured in mg/g, respectively; k₁ (min) and k₂ (g/mg.min) are the PFO and PSO constants, respectively; t (min) is the contact time; n is the order of reaction; k_n,0 is the fractal-like rate constant and h is the fractal-like exponent characterizing whether a system is homogeneous” (Hu et al., 2022; Ngcobo et al., 2025; Haerifar and Azizian, 2014).

q t = q e 1 1 − exp − k 1 t (3)

q t = q e 2 k 2 t k 2 q e t + 1 (4)

q t = q e 1 − k 1 ⁡ exp − k 1 t k 1 + q e k 2 1 − exp − k 1 t (5)

q t = q e 1 − 1 1 + n − 1 k n q e n − 1 t 1 / n − 1 (6)

q t = q e 1 − exp − k 1 , 0 1 − h t 1 − h (7)

q t = q e 1 − 1 1 + q e k 2 , 0 1 − h t 1 − h (8)

2.3.2 Batch adsorption isotherm models

Similarly, adsorption isotherm studies were conducted by comparing two-parameter and three-parameter nonlinear adsorption isotherm models, viz., Freundlich model (Equation 9), Langmuir model (Equation 10), and Langmuir-Freundlich model (Equation 11). where “K_F is the Freundlich constant [(mg/g)/(mg/L)^nF]; n_F is the dimensionless Freundlich’s constant representing surface heterogeneity; Q_L (mg/g) and K_L (L/mg) are the Langmuir maximum monolayer adsorption capacity and the Langmuir equilibrium constant, respectively; Q_LF (mg/g), K_LF (L/mg) and n_LF are the Langmuir-Freundlich maximum adsorption capacity of adsorbate, equilibrium constant, and dimensionless exponent” (Khumalo et al., 2024).

q e = K F C e n F (9)

q e = Q L K L C e 1 + K L C e (10)

q e = Q LF K LF C e n LF 1 + K LF C e n LF (11)

2.3.3 Model evaluation

A variety of statistical parameters were adopted in examining adsorption kinetics and isotherms models’ goodness of fit. This include the coefficient of determination, R² (Equation 12), adjusted-R² (Equation 13), and the chi-square test, χ 2 (Equation 14) (Ngcobo et al., 2025). where “q exp is the experimentally measured amount of adsorbate per mass of adsorbent; q ¯ exp is the average of all measured experimental values of the measured amount adsorbate adsorbed per mass of adsorbent; q m o d e l is the predicted value by the fitted model; p is the number of model parameters; n is the number of dataset points” (Ngcobo et al., 2025; Khumalo et al., 2025).

R 2 = ∑ i = 1 n q exp − q ¯ exp 2 − ∑ i = 1 n q exp − q model 2 ∑ i = 1 n q exp − q ¯ exp 2 (12)

adj . R 2 = 1 − 1 − R 2 n − 1 n − p − 1 (13)

χ 2 = ∑ q e , exp − q e , model 2 q e , model (14)

3 Results and discussion

3.1 Adsorbent characterization

3.1.1 Fourier Transform Infrared (FTIR) spectroscopy

Changes in molecular vibrations of functional groups of the BP synthesised adsorbent were studied by employing the FTIR spectra at a wavenumber ranging from 400 cm^-1 to 4,000 cm^-1. Results of the FTIR spectra before and after adsorption are depicted in Figure 1. The most prevalent functional groups in agricultural waste including BPs are hydroxyls, carboxylic, phenolic, aldehydes, ketones, ethers. Before adsorption, the FTIR spectra gave peaks at 3,336 cm^-1 corresponding to strong O-H stretching bonds as well as N-H stretch of amines of polymeric compounds, namely, pectin and lignin that contains alcohol, phenols and carboxylic acid functional groups. The peaks at 2,917 cm^-1 and 2,852 cm^-1 correspond to C-H stretching of alkanes, while the C=O stretching of carboxylic acids can be seen at a peak of 1736 cm^-1. The peak at 1,641 cm^-1 corresponds to the N-H bending vibrations of amines, 1,444 cm^-1 corresponds to O-H bending of acids, 1,376 cm^-1 is C-H bending of crystalline cellulose and hemicellulose or lignin. Peaks at 1,236 cm^-1 and 1,024 cm^-1 cement the presence of C-O stretching of carbonyls and the peak at 895 cm^-1 is attributed to the presence of aromatic hydrogen. The FTIR spectra obtained for the pristine BP adsorbent before adsorption is congruent with the results reported in the literature (Deshmukh et al., 2017; El-Din et al., 2018).

Furthermore, it is apparent from Figure 1 that there were changes in the intensity of functional groups after the sorption of Cu²⁺ and Cr⁶⁺ on the BP model adsorbent from aqueous solutions. The observed intensity changes explicitly indicate that there were interactions between the metal ions and the functional groups of the model adsorbent. Significant intensity changes were observed at a wavenumber range of 1,236 cm^-1, 1,376 cm^-1, 1,614 cm^-1, and 1,736 cm^-1 for both Cu²⁺ and Cr⁶⁺. The change in intensity at the aforementioned wavenumbers, suggest that the carboxylic and amine groups were most active in the uptake of the model heavy metal ions. Similarly, changes in BP-Cu²⁺ peaks intensity were observed at a wavenumber range of 2,852 cm^-1, 2,917 cm^-1, and 3,336 cm^-1, indicating that the uptake of Cu²⁺ was affected by the alkane and hydroxyl functional groups in addition to the carboxylic and amine functional groups. The FTIR spectra result for the present study suggest that the model BP adsorbent demonstrated higher affinity for Cu²⁺ when compared to Cu⁶⁺.

3.1.2 Scanning electron microscopy and energy dispersive X-ray spectroscopy (SEM/EDS)

The scanning electron microscopy (SEM) analysis was conducted to investigate the surface morphological characteristics of the BP adsorbent before and after heavy metal ions adsorption, viz., Cu²⁺ and Cr⁶⁺. As depicted in Figure 2a, the pristine BP surface exhibited a rough and irregular morphology with no clearly defined pores, indicating a heterogeneous texture. Following the sorption of Cu²⁺ (Figure 2b), the adsorbent surface underwent significant morphological alterations, becoming rougher with distinct fractures and microstructural disruptions. Such changes are consistent with the deposition and binding of Cu²⁺ ions onto active sites, which often induce structural stress and surface heterogeneity (Thanthri et al., 2021). In contrast, the adsorption of Cr⁶⁺ (Figure 2c) did not produce drastic morphological transformations compared to Cu²⁺. Instead, the surface appeared more uneven, with grooves and edge distortions, suggesting a relatively less aggressive interaction but still indicative of ion binding and localized surface modification (He et al., 2023). These observations corroborate previous findings that Cu²⁺ tends to cause more pronounced morphological restructuring due to stronger coordination with functional groups, whereas Cr⁶⁺ adsorption results in subtler surface irregularities (Thanthri et al., 2021; He et al., 2023).

Furthermore, the energy dispersion X-ray spectroscopy (EDS) was employed to elucidate the elemental composition of the model BP adsorbent before and after heavy metal biosorption. It can be observed in Figure 2a, the pristine BP surface was primarily composed of carbon (C), oxygen (O), silicon (Si), chlorine (Cl), and potassium (K), reflecting the lignocellulosic nature of the biosorbent material. Post the biosorption of the model heavy metals on the BP biosorbent, notable compositional changes were observed. Specifically, the biosorption of Cu²⁺ and Cr⁶⁺ resulted in a significant reduction in K, Si, and Cl composition, suggesting ion exchange and displacement of native elements by the adsorbed metals. The detection of element Cr in Figure 2c confirms its successful immobilization within the biosorbent matrix, likely through electrostatic interactions and complexation with functional groups embedded in the cell wall structure (Igberase and Mkhize, 2025). For the biosorption of Cu²⁺, the increase in C and concomitant decrease in O composition indicate that cellulose moieties became more exposed, while hydroxyl groups in lignin actively participated in binding Cu²⁺ ions. The findings of the present work aligns with the findings reported by Marković, Gorgievski (Marković et al., 2023) where Cu²⁺ biosorption was facilitated by hydroxyl and carboxyl functionalities, often accompanied by ion exchange with alkali and alkaline earth metals. The decline in K composition further supports the occurrence of ion exchange, a mechanism widely reported in biosorption processes involving lignocellulosic adsorbents (Ca et al., 2023). From the SEM/EDS results, it can be inferred that EDS results corroborate the SEM findings, demonstrating that Cu²⁺ induces more pronounced chemical and structural modifications compared to Cr⁶⁺. Furthermore, the results indicate the dual mechanisms of biosorption, surface complexation and ion exchange underpinning the removal of the model heavy metals by bio-based adsorbents.

3.1.3 X-ray diffraction (XRD) spectrometry

The XRD analysis provides critical insight into the structural characteristics of the raw BP and the modifications induced after the biosorption of Cr⁶⁺ and Cu²⁺ ions from solution. The diffraction pattern of the raw BP (Figure 3) shows a predominantly amorphous structure, consistent with the complex biopolymeric composition of lignocellulosic materials. The broad amorphous halo is attributed to lignin, a highly branched and irregular aromatic polymer that lacks long-range molecular order and therefore does not contribute to sharp crystalline reflections (Isa et al., 2020; Bhardwaj, 2025). This amorphous nature is typical of agricultural residues and is advantageous for biosorption, as it enhances the accessibility of functional groups involved in metal binding. Despite the dominant amorphous phase, several distinct peaks were observed at 2 θ values of 28 °, 40 °, 50 °, 66 °, and 74 °, indicating the presence of crystalline inclusions. These peaks are attributed to the presence of sylvite (KCl), a natural occurring crystalline salt commonly detected in plant-derived adsorbents. The presence of sylvite in BP has been previously reported by Oyewo, Onyango (Oyewo et al., 2016), confirming the mineralogical consistency of the material. Moreover, a characteristic peak at 2 θ = 22 ° was observed, representing the crystallographic plane of native cellulose. This is attributed to the presence of cellulose, the dominant crystalline form in plant cell walls (Isa et al., 2020; Khairiah et al., 2021). The coexistence of amorphous lignin and semi-crystalline cellulose is typical of lignocellulosic biosorbents and influences both their mechanical stability and sorption behaviour.

Following the biosorption of Cr⁶⁺ and Cu²⁺ ions, notable changes were observed in the XRD patterns (Figures 4, 5). The intensity of the crystalline sylvite peaks decreased markedly, suggesting partial dissolution or structural disruption of KCl during metal uptake. This reduction may be attributed to ion exchange or displacement processes occurring at the biomass surface, where metal ions interact with mineral components and functional groups. The attenuation of these peaks therefore supports the involvement of mineral phases in the biosorption mechanism. After the biosorption of Cr⁶⁺ and Cu²⁺, the most prominent peak remained at approximately 22.3° (see Figures 4, 5), reaffirming the persistence of native cellulose as the dominant crystalline component. However, the appearance of weaker peaks at 14.8° and 34.5° indicates subtle rearrangements within the cellulose microfibrils or interactions between the model metal ions and the adsorbent polysaccharide matrix. The observed modification is attributed to the fact that, the metal binding can induce structural perturbations in lignocellulosic materials, particularly through coordination with hydroxyl and carboxyl groups. It is worth noting that for the sorption of Cr⁶⁺, additional peaks emerged at 15.8°, 21.26°, and 34.82° as indicated in Figure 5. Similar peak shifts and intensity variations have been documented in other biosorbent systems following heavy metal uptake, reflecting changes in crystallinity and possible formation of metal-biosorbent complexes (Sharma and Bhalerao, 2018). These structural alterations further support the hypothesis that biosorption involves both surface interactions and deeper modifications within the biopolymer network.

3.1.4 Brunauer–Emmett–Teller (BET)

The nitrogen adsorption-desorption BET analysis gave a surface area of 5.6909 m²/g, a pore volume of 2.605 m³/g, and an average pore diameter of 2.889 nm for the raw BP powder. The modest surface area and low pore volume are characteristic of dense, non-porous raw biomass and contrast sharply with thermally activated BP, for which values as high as 200 m²/g have been reported following pyrolysis at 450 °C (Ameha et al., 2024). The average pore diameter of 2.8894 nm classifies the material as mesoporous according to the International Union of Pure and Applied Chemistry (IUPAC) convention (2–50 nm), a structural feature that facilitates intraparticle diffusion of hydrated metal ions toward interior binding sites. Importantly, the adsorptive capacity of raw BP is not governed primarily by textural porosity but rather by the density of surface functional groups, notably hydroxyl (-OH), carboxyl (-COOH), and amine (-NH₂) moieties which mediate chemisorptive interactions with Cu²⁺ and Cr⁶⁺ ions (Seleman et al., 2023).

3.2 Effect of operating parameters on the biosorption of Cu²⁺ and Cr⁶⁺ on BP

3.2.1 Effect of solution pH

The solution pH is one of the most influential parameters governing the sorption of metal ions onto biomass-derived adsorbent. Available literature (Ngomane et al., 2026; Babel and Kurniawan, 2004; Volesky, 2007; Jahin et al., 2024) suggests that solution pH affects both the surface charge of the adsorbent and the speciation of metal ions in solution, thereby controlling the extent of electrostatic attraction and/or repulsion between the adsorbent and the adsorbate ions. Batch solid-liquid biosorption experiments suggest that the BP model biosorbent used in the current study follows the general principles as reflected in the biosorption pH profile (Figure 6) for Cu²⁺ and Cr⁶⁺ across the studied pH range 2–7. From the experimental results presented in Figure 6, it is apparent that Cu²⁺ biosorption increased sharply from pH 2 to pH 4, reaching a maximum of 97% at pH 4, after which the removal efficiency gradually decreases. The relatively low Cu²⁺ removal efficiency of 3% at pH 2 is attributed to the BP surface being highly protonated due to excess H⁺ ions. As such, the competition between H⁺ and Cu²⁺ for binding sites was relatively intense resulting in extremely low biosorption removals of 3% for Cu²⁺. According to an early study by Ho and McKay (Ho and McKay, 1999) this behaviour is consistent with typical biosorbents containing carboxyl, hydroxyl, and amino groups. On the other hand, at moderate pH of 3–4, the protonation on the BP surface decreases, exposing negatively charged functional groups such as COO⁻ and -OH^- resulting in an increase in electrostatic attraction between Cu²⁺ and the deprotonated BP sites. As such, the Cu²⁺ removal of 97% at a solution pH of 4 can be attributed to an increase in electrostatic attraction forces. Similarly, studies by Annadurai, Juang (Annadurai et al., 2003) and Nyairo, Njewa (Nyairo et al., 2025) reported an optimal pH of 4 for the uptake of Cu²⁺ on fruit-waste biosorbent. The observed pH profile for the biosorption of Cu²⁺ on BP is congruent to studies reported in the literature. It is worth noting that, at a solution pH range of 5–7, there was a gradual decrease in the biosorption of Cu²⁺. According to Ahalya, Ramachandra (Ahalya et al., 2003) the observed trend is attributed to the partial precipitation of Cu(OH)₂ above pH 5 which reduces the availability of free Cu²⁺ ions. Furthermore, an increase in solution pH above 4 results in a competition between hydrolysed species (i.e., CuOH⁺, Cu(OH)₂) and biosorption sites. From the results obtained, it can be inferred that, the biosorption behaviour of Cu²⁺ on BP, is typical of divalent cations on lignocellulosic biomass, i.e., low at strong acidic pH, maximum at mildly acidic pH, and decreasing at near-neutral pH. A one-way analysis of variance (ANOVA) conducted at 95% confidence level yielded an F-value of 20.53, which substantially exceeded the critical threshold of F-critical (4.96), with an associated p-value of 1.09 × 10⁻³ (p < 0.05). The ANOVA results provide strong statistical evidence that solution pH exerted a significant effect on Cu²⁺ percentage removal, confirming that the observed variations in removal across pH levels were not attributable to random experimental error. The magnitude of the F-ratio relative to the critical value further confirms the robustness of this pH-dependent response, as corroborated by the distinct separation evident in the corresponding percentage removal error bars.

Furthermore, from Figure 6, it can be observed that Cr⁶⁺ biosorption increased from pH 2 to pH 4, reaching a maximum of 65% at pH 4, followed by a steady decline toward neutral pH conditions. It is worth noting that, Cr⁶⁺ exists primarily as HCr O 4 −, C r 2 O 7 2 −, and Cr O 4 2 − depending on solution pH (Nyairo et al., 2025; Park et al., 2010). The BP surface is positively charged at a low pH of 2 due to protonation, favouring electrostatic attraction with anionic Cr⁶⁺ species. However, the uptake of Cr⁶⁺ was relatively low recording a removal of 11%. The observed trend is attributed to the domination of HCr O 4 − at pH 2 thus compromising the uptake of Cr⁶⁺ (Nyairo et al., 2025). It is imperative to note some functional groups may be overly protonated, reducing binding capacity. At moderate pH range of 3–4, the surface remains positively charged but less excessively protonated increasing the electrostatic attraction between the biosorbent surface and Cr⁶⁺ anions becomes more favourable reaching a maximum biosorption of 65% at pH 4 as evident from Figure 6. The findings of the present work is congruent with the work reported by Karthikeyan, Rajgopal (Karthikeyan et al., 2005) and Paranjape and Sadgir (Paranjape and Sadgir, 2023) on the biosorption of Cr⁶⁺ on plant-based adsorbents. Similarly to the biosorption of Cu²⁺, at a higher pH range of 5–7, the biosorbent surface became increasingly deprotonated and negatively charged increasing the electrostatic repulsion between the biosorbent surface and Cr ions existing as C r 2 O 7 2 −, and Cr O 4 2 − thus compromising the uptake rate (Nyairo et al., 2025) recording low uptake percentage of 30%. The effect of solution pH on the uptake of Cu²⁺ and Cr⁶⁺ on BP as a biosorbent is summarised in Table 1 with the associated mechanism as supported in the literature (Nyairo et al., 2025; Jangir et al., 2024). A comparable trend was observed for Cr⁶⁺ removal, wherein a one-way ANOVA, at 95% confidence level returned an F-value of 18.71, again exceeding the critical threshold of F-critical = 4.96, with a corresponding p-value of 1.5 × 10⁻³ (p < 0.05). Consistent with the findings for Cu²⁺, these statistical parameters confirm that solution pH exerted a statistically significant influence on Cr⁶⁺ percentage removal, with the observed inter-group variation being attributable to the effect of pH rather than experimental error. Collectively, the ANOVA results for both metal ions demonstrate that pH is a critical determinant of biosorption performance across the system.

TABLE 1. Comparative interpretation on the biosorption of Cu²⁺ and Cr⁶⁺ with variation in solution pH.

3.2.2 Effect of contact

In a typical solid-liquid adsorption process, contact time is a critical process operating parameter influencing the rate and extent of metal uptake during biosorption. It determines how quickly the biosorbent surface interacts with metal ions and how long it takes to reach equilibrium. Figure 7 presents findings of the present study on the effect of contact time on Cu²⁺ and Cr⁶⁺ percentage removal. From the results presented in Figure 7, it can be observed that the removal of Cu²⁺ increased rapidly within the first 40 min, rising from 0% at time 0 min to 87% at 40 min. This initial rapid uptake is attributed to the abundance of available active sites on the BP surface, i.e., carboxyl, hydroxyl, and amino functional groups, confirmed from the FTIR and SEM/EDS results in Figures 1 and 2. As such, this suggest that, the fast initial biosorption phase is defined by surface biosorption, where model metal ions readily bind to exposed functional groups. Between 40 min and 80 min, Cu²⁺ removal increased more gradually, reaching 92%. This slower phase indicates the onset of intraparticle diffusion, where metal ions migrate into the internal pores of the biosorbent (Danish et al., 2022). After 80 min, the removal efficiency stabilised at approximately 90%–92%, suggesting that equilibrium had been reached as indicated by the plateau. The relatively high percentage removal of up to 92% demonstrates the strong affinity of BP biosorbent for Cu²⁺, the findings of the study are consistent with the work reported by Jahin, Khedr (Jahin et al., 2024) on BP biosorbent indicating its effectiveness as a low-cost, sustainable biosorbent for divalent metal ions.

Similarly, the uptake of Cr⁶⁺ increased with contact time (Figure 7), though at a slower rate compared to Cu²⁺. The removal moved from 0% to 40% within 5 min, indicating rapid initial biosorption due to electrostatic attraction between protonated BP surface and anionic Cr⁶⁺ species, i.e., HCr O 4 − and Cr 2 O 7 2 − (Danish et al., 2022). The observed removal trend is consistent with the literature, demonstrating that BP biosorbents possess positively charged sites under acidic conditions that favour Cr⁶⁺ uptake. Notable, after 15 min, Cr⁶⁺ removal reached 45%, followed by a gradual increase to 54%–56% between 80 min and 140 min. The slower uptake compared to Cu²⁺ was expected due to Cr⁶⁺ existing predominantly as anions, which compete with other anions in solution and experience repulsion as the biosorbent surface becomes less protonated over time. This mechanism is not explicitly accounted for in the present study, however, it has been accounted for in the literature (He et al., 2023). From the results presented in Figure 7, it can be inferred that the equilibrium state for Cr⁶⁺ appears to be reached around 100–140 min, with percentage removal stabilising at 55%–56%. This aligns with previously done studies showing that Cr⁶⁺ biosorption on plant-based adsorbents typically reaches equilibrium later than Cu²⁺ due to slower diffusion and weaker binding interactions (Jangir et al., 2024). In contrast to the pH-dependent responses, one-way ANOVA analysis of the effect of contact time on metal biosorption yielded non-significant results for both Cu²⁺ and Cr⁶⁺ at 95% confidence level. An F-value of 0.577 was recorded for Cu²⁺ biosorption, falling well below the critical threshold of F-critical = 4.35, with a p-value of 0.456 (p > 0.05). Similarly, Cr⁶⁺ biosorption recorded an F-value of 0.301, also below the critical value of 4.35, with a corresponding p-value of 0.590 (p > 0.05). The failure to reject the null hypothesis in both cases indicates that, beyond the initial rapid uptake phase, progressive increases in contact time did not produce statistically significant changes in Cu²⁺ or Cr⁶⁺ removal from solution. This is consistent with the attainment of adsorption equilibrium, as visually corroborated by the plateau regions evident in the biosorption profile presented in Figure 7.

3.2.3 Biosorbent dosage

Increasing the mass of a biosorbent typically enhances the availability of active binding sites, thereby improving metal uptake until equilibrium or saturation conditions are reached. Herein, Figure 8 depicts the findings of the study on the effect of biosorbent dosage in Cu²⁺ and Cr⁶⁺ uptake on BP as a model biosorbent. Cu²⁺ removal increased markedly with increasing biosorbent dosage, rising from 17% at 1 g/L to 91% at 6 g/L. The steep increase between 1 g/L and 4 g/L reflects the rapid availability of additional functional groups, i.e., carboxyl, hydroxyl, and amino groups on BP (see Figure 1) which are reported in literature to bind divalent metal ions through complexation and electrostatic attraction (Jahin et al., 2024; Jangir et al., 2024). Beyond a biosorbent dosage of 4 g/L, the increase in Cu²⁺ removal became less pronounced, reaching a plateau at 91%. This behaviour is a characteristic of biosorption systems where, after a certain dosage, the available metal ions become the limiting factor rather than the number of active sites. Similarly, Jangir et al. (2024) reported that excess adsorbent mass leads to particle aggregation and reduced surface accessibility, thus compromising the adsorbate uptake from solution.

Cr⁶⁺ removal also increased with biosorbent dosage, though less dramatical than Cu²⁺. Removal rose from 58% at 1 g/L to 65% at 4 g/L, followed by a slight fluctuation between 63% and 64% at higher dosages. This moderate increase is attributed to the anionic nature of Cr⁶⁺ species, i.e., HCr O 4 − and Cr 2 O 7 2 −, which rely primarily on electrostatic attraction to protonated sites on the biosorbent surface (Ameha et al., 2024). Available literature (Ameha et al., 2024) suggests that, BP have shown to remove Cr⁶⁺ effectively through the aforementioned mechanisms, however, the efficiency is typically lower than for cationic metals due to competition with other anions and repulsion at higher pH values. From Figure 8, it can be inferred that a biosorbent dose of 4 g/L gave the highest Cr⁶⁺ percentage removal of 65%. In comparison between Cu²⁺ and Cr⁶⁺ uptake on BP as a model biosorbent, the results suggest that Cu²⁺ removal is more sensitive to biosorbent dosage than Cr⁶⁺. The observed trend can be attributed to several mechanistic factors. Cu²⁺ binds strongly to negatively charged functional groups through complexation and ion exchange. On the other hand, Cr⁶⁺ being anionic, depends on electrostatic attraction to protonated sites, which are limited and more pH dependent. The results obtained suggest that the model biosorbent exhibit higher affinity for divalent cations than for oxyanions as confirmed in the FTIR results in Figure 1 and from the work reported by Jahin, Khedr (Jahin et al., 2024). Furthermore, Cu²⁺ removal was observed to increase up to a biosorbent dosage of 6 g/L, while Cr⁶⁺ removal plateaus earlier, indicating stronger and more numerous binding interactions for Cu²⁺. One-way ANOVA analysis further demonstrates that biosorbent loading exerted a statistically significant effect on the removal of both metal ions. p-values of 4.69 × 10⁻⁴ and 1.64 × 10⁻¹² were recorded for Cr⁶⁺ and Cu²⁺, respectively, both falling substantially below the 0.05 significance threshold at the 95% confidence level. The rejection of the null hypothesis for both ions confirms that observed variations in Cu²⁺ and Cr⁶⁺ uptake across different BP biosorbent doses were statistically significant and not attributable to random experimental variation. Notably, the considerably lower p-value recorded for Cu²⁺ (1.64 × 10⁻¹²) relative to Cr⁶⁺ (4.69 × 10⁻⁴) suggests a stronger dose-dependent biosorption response for Cu²⁺, indicating greater sensitivity of Cu²⁺ removal efficiency to changes in biosorbent loading under the experimental conditions employed.

3.2.4 Adsorbate initial concentration

From the results presented in Figure 9, the removal performance of BP biosorbent toward Cu²⁺ and Cr⁶⁺ exhibited a clear dependence on the initial ion concentration, reflecting the interplay between surface functional group availability, mass-transfer driving forces, and competitive adsorption phenomena. From the results obtained, it is apparent Cu²⁺ removal efficiencies ranged from 83% to 95%, while Cr⁶⁺ removal efficiencies were comparatively lower at 36%–65% across the tested concentration range (5–40 mg/L). The divergence in performance can be attributed to BP exhibiting a higher affinity for divalent cations than oxyanion-forming species, i.e., Cr⁶⁺, owing to differences in ionic radius, hydration energy, and electrostatic interactions with surface ligands (Jahin et al., 2024; Miretzky and Cirelli, 2010; Hossain et al., 2012). Similar Cu²⁺ removal increased sharply from 83% at 5 mg/L to 95% between 10–20 mg/L, indicating that the initial rise in concentration enhanced the mass-transfer driving force, thereby improving diffusion of Cu²⁺ ions toward active sites. The plateau observed between 10–20 mg/L suggests that the available binding sites were sufficient to accommodate the increased ion load without saturation. However, at concentrations above 30 mg/L, Cu²⁺ removal declined slightly to 92% and further to 84% at 40 mg/L, indicating the onset of site saturation. The decline can be attributed to an increase in intra-particle diffusion resistance at higher concentrations, a phenomenon widely reported by Weber Jr and Morris (Weber and Morris, 1963).

In contrast, Cr⁶⁺ removal exhibited a decreasing trend with increasing concentration, dropping from 65% at 5 mg/L to 36% at 40 mg/L. As such, this uptake profile suggests that the biosorption of Cr⁶⁺ is more sensitive to site saturation and electrostatic repulsion effects. It is imperative to note that, the results obtained suggest that the biosorption of Cr⁶⁺ typically exists as negatively charged oxyanions (CrO 4 2 −; HCrO 4 −) in aqueous media, which interact less favourably with predominantly negatively charged functional groups on untreated BP surface (Mamo et al., 2025). Therefore, as the concentration increases, competition among oxyanions intensifies, leading to reduced biosorption efficiency. Moreover, Cr⁶⁺ biosorption often requires reduction to Cr³⁺ prior to binding, a process mediated by surface phenolics or residual sugar (Gupta et al., 2025). At relatively higher concentrations, the reduction step becomes rate-limiting, contributing to the observed decline in the percentage removal. Therefore, the relatively high decrease in Cr⁶⁺ removal compared to Cu²⁺ is attributed to the mechanistic differences between cationic and anionic adsorption on BP surface. Furthermore, the Cu²⁺ system recorded a p-value of 2.56 × 10⁻⁷ while the Cr⁶⁺ system recorded a p-value of 1.28 × 10⁻³, suggesting that there was a strong variation in the percentage removal of Cu²⁺ and Cr⁶⁺ with variation in initial concentration of adsorbate.

3.2.5 Biosorption kinetics studies

The kinetic behaviour of Cr⁶⁺ and Cu²⁺ biosorption onto BP biosorbent demonstrates distinct mechanistic pathways that reflect both the physicochemical nature of the metal ions and the heterogeneous surface characteristics of the biosorbent. From the results presented in Table 2, it is apparent that across all models evaluated, i.e., PFO, PSO, F-PFO, F-PSO, MOE, and PNO model, substantial differences emerged between Cr⁶⁺ and Cu²⁺, accentuating the importance of ion-specific interactions and surface energy distribution.

TABLE 2. Cr⁶⁺ and Cu²⁺ biosorption kinetics parameters on BP biosorbent.

3.2.5.1 Classical kinetic models: PFO versus PSO

For both Cr⁶⁺ and Cu²⁺, the PFO and PSO models provided relatively high R² values > 0.95, indicating that both physisorption and chemisorption may contribute to the overall uptake process. However, the PFO model yielded superior model fit for Cr⁶⁺ (R² = 0.9930, χ 2 = 0.0193) compared to Cu²⁺ (R² = 0.9776, χ 2 = 0.0388), suggesting that Cr⁶⁺ biosorption is dominated by diffusion controlled and/or weakly bound interactions (Park et al., 2010; Babel and Kurniawan, 2003). In contrast, the PSO model produced higher equilibrium capacities for both ions, particularly Cu²⁺, q_e = 14.092 mg/g, and maintained strong statistical performance, i.e., R² = 0.9546 and χ 2 = 0.0785. The results obtained can be attributed to the tendency of divalent cations to form stronger inner-sphere complexes with oxygenated functional groups on the biosorbent surface as discussed by Anastopoulos and Kyzas (Anastopoulos and Kyzas, 2016) in their review. The higher PSO rate constant for Cr⁶⁺, i.e., k₂ = 0.3362 g/(mg.min) compared to Cu²⁺, i.e., k₂ = 4.9 × 10⁻³ g/(mg.min) further indicates faster chemisorption kinetics for Cr⁶⁺, despite its lower overall capacity.

3.2.5.2 Fractal kinetic models

The incorporation of fractal parameters in the F-PFO and F-PSO models significantly improved the goodness of fit for Cr⁶⁺, with F-PFO yielding the highest R² (0.9960) and lowest χ 2 (0.0111) among all models. The fractal exponent h = 0.8483 indicates moderate surface heterogeneity and time-dependent sorption energy distributions, consistent with the irregular pore structure and heterogeneous functional groups of BP-derived biosorbent as cemented by the SEM and FTIR results. For Cu²⁺, however, the fractal models did not outperform the classical models. The F-PFO model demonstrated a reduced fit quality, i.e., R² = 0.9676, χ 2 = 0.0866, and the fractal exponent was notably low, i.e., h = 0.1401, suggesting that Cu²⁺ adsorption proceeds through more uniform and energetically consistent binding sites. The results obtained are attributed to the typical behaviour of divalent cations interacting with carboxyl, hydroxyl, and phenolic groups in a more predictable manner as discussed by Yang et al. (2026). The F-PSO model provided a marginal improvement over PSO for Cr⁶⁺, i.e., R² = 0.9985, and R² = 0.9952, respectively, but not for Cu²⁺, where the adj. R² dropped to 0.9313. The fractal exponent for Cu²⁺ (i.e., h = 0.0397) again indicated minimal fractal behaviour, suggesting that, Cu²⁺ adsorption is less influenced by surface irregularities.

3.2.5.3 Mixed-order and power-law models

The MOE and PNO models provided additional insight into the complexity of the adsorption process. The MOE model, which integrates first- and second-order contributions, performed comparably to PSO for both ions, with identical R² values for Cr⁶⁺ (0.9952) and Cu²⁺ (0.9546). This suggests that both diffusion-driven and chemisorption mechanisms operate simultaneously, a behaviour commonly observed in biosorption systems with multi-site interactions (Largitte and Pasquier, 2016). The PNO model yielded strong fits for both Cr⁶⁺ (R² = 0.9955; χ 2 = 0.0126) and Cu²⁺ (R² = 0.9591; χ 2 = 0.0706), with reaction orders n > 1, indicating complex, non-integer kinetics. The results suggest that, the investigated systems’ sorption rates vary with surface coverage and multiple parallel pathways contribute to the overall uptake (Hubbe et al., 2019). The higher reaction order for Cr⁶⁺ (n = 2.3709) suggests stronger deviation from classical kinetics, consistent with its multivalent speciation and electrostatic sensitivity.

3.2.5.4 Comparative mechanistic interpretation

From the adsorption kinetics results presented in Table 1, it can be inferred that, Cr⁶⁺ adsorption is strongly influenced by surface heterogeneity, as evidenced by the superior performance of fractal models and higher fractal exponents. This behaviour is in line with the multi-speciation of Cr⁶⁺ (i.e., HCrO 4 −; Cr 2 O 7 2 −) and its sensitivity to microenvironmental variations (Zhu et al., 2025). On the other hand, Cu²⁺ biosorption follows more uniform chemisorption pathways, with PSO and MOE models providing the most reliable fits. The higher equilibrium values reflect stronger affinity for the model biosorbent functional groups, the findings of the study are congruent with established trends for divalent transition metals reported in the literature (Anastopoulos and Kyzas, 2016). It is imperative to note that, fractal models are particularly valuable for systems where surface roughness and energetic heterogeneity dominate, as in the case of Cr⁶⁺, however, they may offer limited benefit for ions with more predictable binding behaviour. Furthermore, the PNO model confirms the presence of multi-pathway kinetics, supporting the interpretation that both ions undergo complex biosorption processes that cannot be fully capture by classical models alone. The findings of the present study indicate the importance of selecting kinetic models that reflect the physicochemical nature of both the biosorbent and the adsorbate. It can be inferred that, for Cr⁶⁺, models incorporating fractal behaviour or non-integer reaction orders provide more accurate mechanistic insight, while for Cu²⁺, classical chemisorption models remain adequate. This distinction is crucial for optimizing biosorbent modification strategies, predicting performance in dynamic systems, and scaling up adsorption processes for industrial wastewater treatment.

3.2.5.5 Comparison with literature studies

From the studies presented in Table 2, the PSO model clearly dominates as the primary kinetic descriptor of heavy metals biosorption on lignocellulosic waste (Jahin et al., 2024; Annadurai et al., 2003; Huang et al., 2024; Raji et al., 2023). This is consistently interpreted as evidence of chemisorption involving valence forces or electron sharing between metal ions and surface functional groups such as -OH, -COOH, and phenolic moieties. In this context, the present study’s findings that Cu²⁺ kinetics on raw BP are best captured by PSO and MOE, with high equilibrium capacities and strong statistical indicators, aligns closely with the behaviour reported for Cu²⁺ on multi-metal systems on mixed fruit-peel biochar (Raji et al., 2023), and Zn²⁺ on activated BP (Jahin et al., 2024). For Cr⁶⁺, the PSO model is also frequently reported as the best-fit model on modified lignocellulosic adsorbents, including acid-modified BP (Huang et al., 2024) where chemisorption and in some cases, redox processes are invoked. In contrast, the present study shows that the F-PFO and F-PSO models, together with PNO model, provide an even better description of Cr⁶⁺ kinetics than the classical PSO model, indicating the importance of surface heterogeneity and non-integer reaction orders that are not explicitly considered in most of the recent works.

From available literature (Mohammed and Hameed, 2017), it can be noted that surface heterogeneity is acknowledged through the use of intra-particle diffusion, however, they generally stop at classical PFO/PSO model formalisms. It is worth noting that a number of studies (Jahin et al., 2024; Annadurai et al., 2003; Huang et al., 2024; Raji et al., 2023; Mohammed and Hameed, 2017) have reported the classical PSO as the primary kinetic model, with intra-particle diffusion indicating heterogeneous energy distributions and multi-stage uptake. However, these models do not explicitly quantify the fractal nature of the surface or the time-dependent rate coefficients. The present study advances this by explicitly applying fractal-like PFO and PSO and a power-law PNO model to Cr⁶⁺ and Cu³⁺ biosorption on raw BP. For Cr⁶⁺, the superior performance of F-PFO and F-PSO, together with non-integer reaction orders from PNO, indicates that biosorption occurs on a highly heterogeneous energy landscape, where local microenvironments and pore structures strongly modulate the rate. This mechanistic nuance is only indirectly inferred in the work reported by Mohammed and Hameed (Mohammed and Hameed, 2017) where intra-particle diffusion is invoked but not linked to explicit fractal parameters. For Cu²⁺, the present work demonstrates that fractal models offer only marginal improvement over PSO and MOE, suggesting that Cu²⁺ interacts preferentially with a subset of more uniform sites behaviour similar to that observed for Pb²⁺ (Raji et al., 2023) and Zn²⁺, where monolayer chemisorption on relatively homogeneous surfaces is proposed.

A key distinction between the present study and many of the comparative works presented in Table 3 lies in the degree of surface modification. Studies (Ameha et al., 2024; Jahin et al., 2024; Huang et al., 2024) reported in the literature use chemically and/or thermally activated biosorbents, which typically exhibit higher surface area, more uniform pore structures, and increased densities of oxygenated functional groups. This often leads to more uniform adsorption sites and simpler kinetic behaviour that can be adequately described by PSO and occasionally the PFO classical models without resorting to fractal or distributed-order models. In contrast, the present study uses raw BP, preserving a broader distribution of pore sizes, surface roughness, and functional group environments. The fact that the F-PFO and F-PSO models outperform classical models for Cr⁶⁺ is therefore consistent with a more rugged, heterogeneous surface where adsorption occurs over multiple time scales and local energy barriers. This contrasts with the activated BP system for Cr⁶⁺ reported by Ameha et al. (2024), where PFO alone is sufficient, and with acid-modified BP system reported by Huang, Campbell (Huang et al., 2024) where PSO dominates, both reflecting a “smoothing” of energy landscape through activation. This comparison suggests a practical design principle, surface modification not only enhances capacity but also simplifies kinetic behaviour, making classical models more adequate. When biosorbents are used without any modification, advanced kinetic formalisms such as fractal and PNO models become more informative for mechanistic interpretation and process design.

TABLE 3. Literature studies on biosorption kinetics.

Studies reported in the literature on heavy metals biosorption are presented without deeper exploration of how surface heterogeneity differentially affects each ion. Herein, the stronger sensitivity of Cr⁶⁺ to surface heterogeneity, as evidenced by clear advantage of fractal and PNO models, contrasts with more classical PSO-type behaviour of Cu²⁺. This mirrors, to some extent, the distinction observed in the work reported by Jahin et al. (2024), where Cu²⁺ and Zn²⁺ on activated BP follow different kinetic models (i.e., PFO versus PSO). The present study, however, formalises this ion-specific heterogeneity through explicit fractal parameters and non-integer reaction orders, rather than relying solely on qualitative interpretation.

3.3 Biosorption isotherms

The biosorption behaviour of Cu²⁺ and Cr⁶⁺ ions on BP was evaluated using three equilibrium models, viz., Freundlich, Langmuir, and Langmuir-Freundlich as presented in Table 4. The results obtained shows distinct biosorption mechanisms for each metal ion, reflecting the heterogeneous surface chemistry of BP as a model biosorbent. The Freundlich isotherm model provided an excellent fit (see Figure 10a) for Cr⁶⁺ biosorption (R² = 0.983, RMSE = 0.873) with K_F = 0.855 (mg/g) (L/mg)^1/0.557 and n_F = 0.557. The value of n_F < 1 indicates favourable adsorption on heterogeneous sites, consistent with multilayer uptake facilitated by hydroxyl and carboxyl groups in lignocellulosic biomass (Jahin et al., 2024). In contrast, Cu²⁺ biosorption demonstrated weaker conformity recording R² and RMSE values of 0.7517 and 6.117, respectively, despite a higher K_F value of 5.5684. The relatively low n_F value of 0.2524 suggests strong heterogeneity and possible clustering of Cu²⁺ ions, which may reduce predictive accuracy of the Freundlich model.

TABLE 4. Biosorption isotherms parameters for the biosorption of Cr⁶⁺ and Cu²⁺ on BP biosorbent.

Furthermore, the Langmuir model described Cu²⁺ adsorption more effectively recording R² and adj. R² values of 0.9000 and 0.8667, respectively, with a maximum monolayer capacity of Q_L = 13.51 mg/g and a high affinity constant, K_L = 0.6128 L/mg. This indicates strong binding affinity of Cu²⁺ to BP functional groups, likely through ion exchange with carboxyl moieties (Oladipo et al., 2019). Similarly, for the biosorption of Cr⁶⁺, the Langmuir isotherm model also provided a good fit, i.e., R² = 0.9632, with Q_L = 11.39 mg/g, though slightly less accurate than the Freundlich isotherm model. This suggest that Cr⁶⁺ biosorption involves both monolayer and heterogeneous mechanisms. On the other hand, the Langmuir-Freundlich isotherm model, offered the most balanced description of adsorption behaviour. For the Cr⁶⁺ system, the model recorded an R² value of 0.9814 with a notably high biosorption capacity of Q_LF = 37.79 mg/g, indicating potential multilayer biosorption and strong site heterogeneity. For Cu²⁺, the model also performed relatively well recording an R² value of 0.9212, with Q_LF = 12.47 mg/g, and n_LF = 1.59, suggesting cooperative biosorption where binding of one ion enhances subsequent uptake (Howe et al., 2014). As such, this duality reflects the complex interplay of electrostatic attraction, ion exchange, and surface heterogeneity in biosorption processes. From the isotherms results presented in Table 4, it can be inferred that, Cr⁶⁺ biosorption on BP is better represented by the Freundlich and Langmuir-Freundlich isotherms, both of which explicitly account for surface heterogeneity. The excellent fit and high apparent capacity by the Langmuir-Freundlich isotherm model suggest that Cr⁶⁺ interacts with a spectrum of sites, including high-energy domains that become increasingly important at higher equilibrium concentrations. This is attributed to available literature (Afolabi et al., 2021) suggesting that, Cr⁶⁺ biosorption on BP and other lignocellulosic biomass involves a combination of electrostatic attraction, reduction to Cr³⁺, and subsequent complexation, leading to non-uniform site occupation and layered deposition. While Cu²⁺ biosorption aligns more closely with the Langmuir-Freundlich model, providing the best compromise between physical interpretability and statistical performance. The cooperative parameter, i.e., n_LF > 1 indicates that once initial Cu²⁺ binding occurs, local structural rearrangements or change compensation effects facilitate further uptake, a phenomenon also observed in bi-solute systems where Cu²⁺ competes or co-adsorbs with other metals on BP (Afolabi et al., 2021). The higher monolayer capacity for Cu²⁺ (i.e., 13.5 mg/g) compared to Cr⁶⁺ (i.e., 11.4 mg/g) under Langmuir isotherm suggests the model biosorbent is effective for Cu²⁺ remediation. These findings corroborate the work reported by Jahin et al. (2024) cementing BP as a sustainable, low-cost biosorbent for heavy metal removal from wastewater.

3.3.1 Comparison with literature

The biosorption behaviour observed in the present study demonstrates important consistencies and distinctions when positioned within recent literature (2018–2026) as presented in Table 5 on Cr⁶⁺ and Cu²⁺ sorption using agro-waste derived biosorbents. For Cr⁶⁺, the Freundlich model provide the best statistical fit (i.e., R² = 0.983, adj. R² = 0.977, and χ 2 = 0.017), indicating heterogeneous multilayer adsorption. The results obtained are in congruence with the findings reported for activated rice straw (Putra et al., 2022) and corncob biochar (Nguyen et al., 2024), where surface heterogeneity and non-uniform binding energies governed Cr⁶⁺ uptake. It is imperative to note that, the work reported by Hu et al. (2019) reported higher monolayer capacities for Cr⁶⁺ of up to ∼93mg/g using engineered corn stalk biochar. The engineered corn stalk biochar system involves relatively high pyrolysis temperatures and/or chemical modification, leading to increased surface area and enriched oxygen-containing functional groups. In contrast, the moderate Langmuir capacity obtained in the present work of 12.4676 mg/g suggests that the biosorption efficiency is strongly dependent on surface chemistry and activation strategy, yet the superior Freundlich fit confirms that biosorption does not occur on energetically uniform sites. The Langmuir-Freundlich model also provided competitive fitting, i.e., R² = 0.981, cementing the coexistence of monolayer biosorption and site heterogeneity, which is consistent with the work reported by Nguyen, Ly-Tran (Nguyen et al., 2024). Herein, for Cu²⁺, the Langmuir-Freundlich model yielded the best performance indicating heterogeneous monolayer biosorption with cooperative binding characteristics. This finding is comparable to dual-metal adsorption reported by Hu et al. (2019) and to surface complexation behaviour as discussed by Bayuo (2021), where Cu²⁺ uptake was predominantly attributed to inner-sphere complexation and ion exchange with -COOH and -OH groups. Although higher Cu²⁺ adsorption equilibrium capacities of up to 105 mg/g (Hu et al., 2019) have been reported in modified biochar, such systems frequently rely on chemical impregnation or composite formation. The present study, while yielding a moderate Langmuir-Freundlich equilibrium capacity of 12.4676 mg/g, demonstrates statistically rigorous model discrimination using multiple error functions, viz., R², adj. R², χ 2, and RMSE, which remains underutilized in many adsorption studies that rely solely on linearised R² values. Importantly, the distinct modelling behaviour between Cr⁶⁺ (Freundlich-dominant) and Cu²⁺ (Langmuir-Freundlich dominant) within the same biosorbent system provides mechanistic differentiation. It is apparent that, Cr⁶⁺ removal likely involves electrostatic attraction followed by partial reduction to Cr³⁺ and subsequent binding, whereas Cu²⁺ removal is governed primarily by coordination and ion exchange without redox transformation. From the results obtained, it can be inferred that the present study offers a statistically robust isotherm validation and clear mechanistic discrimination between anionic (Cr⁶⁺) and cationic (Cu²⁺) species on an agro-waste-derived material.

TABLE 5. Biosorption studies reported in the literature.

3.4 Implications for real wastewater treatment

Generally, real wastewater is characterized by multiple competing ions, variable pH, suspended solids, and organic matter that can block activate sites or alter metal speciation (Simate et al., 2011). Studies applying BP biosorbents to real effluents show that performance typically decreases relative to synthetic solutions because of ionic competition and surface fouling. Jahin et al. (2024) demonstrated successful removal of multiple metals from real wastewater using chemically activated BP, but with lower biosorption capacities than in controlled batch tests, cementing the need for matrix-specific optimization. In the case of Cr⁶⁺, the presence of organic matter and reducing agents can partially convert Cr⁶⁺ to Cr³⁺, altering uptake mechanisms. Cu²⁺, being more stable, competes strongly with other divalent ions, viz, Ca²⁺, Mg²⁺, and Zn²⁺ which may reduce its affinity for BP functional groups. Moreover, in the context of selectivity and competitive adsorption, real wastewater rarely contains a single metal. The selectivity of BP becomes critical. As per the findings of the present study, Cu²⁺ often shows higher removal efficiency than Cr⁶⁺ even in mixed systems due to stronger complexation with carboxyl, hydroxyl, and amino groups.

BP biosorbents have demonstrated to be attractive for low-cost treatment, on the basis that they are agricultural waste, however real wastewater applications require addressing few challenges, viz., biosorbent regeneration using concentrated acid desorption, biosorbent stability in terms of lifespan, and process integration to the currently existing treatment works. It is imperative to note that, these challenges are not accounted for in the present work. Cr⁶⁺ is characterised as carcinogenic and strictly regulate in most parts of the globe, while Cu²⁺ has toxicity thresholds for aquatic life. The ability of BP to remove both Cr⁶⁺ and Cu²⁺ even at moderate efficiencies, positions them as sustainable, low-cost alternatives for pre-treatment or polishing stages in industrial wastewater systems. However, for regulatory compliance, especially for Cr⁶⁺, biosorption alone may not achieve discharge limits. Integration with other technologies such as membrane polishing may be required. Furthermore, BP are abundant agricultural waste, making them ideal for circular-economy models. Their use in real wastewater treatment supports, solid-waste valorisation, low-carbon remediation technologies, and community-level treatment systems in developing regions. Yet, large-scale deployment requires, standardisation of biosorbent preparation, consistent performance across wastewater matrix, and life-cycle assessment to quantify environmental benefits. These requirements are not explicitly accounted for in the present work.

4 Study limitations, and future research directions

Most biosorption studies including the present study in the use of BP as a biosorbent are conducted in batch mode, which provides controlled conditions for evaluating kinetics, isotherms, and mechanistic pathways. However, batch systems do not replicate the hydrodynamic, mass-transfer, and breakthrough dynamics encountered in continuous treatment units. The absence of column experiments for the present work limits understanding of 1) flow-dependent mass transfer, which can significantly reduce effective uptake compared to batch equilibrium; 2) bed clogging and compaction, especially for soft, fibrous biomaterials like BP; 3) breakthrough behaviour, which determines operational lifespan and regeneration frequency; and 4) scale-up feasibility, since industrial wastewater treatment relies primarily on fixed-bed systems. For future work, continuous-flow column studies should be conducted to quantify breakthrough curves, evaluating the influence of flow rate and bed depth. Studies incorporating granulated or pelletised BP derivatives may improve mechanical stability and reduce pressure drop, enabling more realistic pilot-scale testing.

Herein, biosorbent regeneration studies were not conducted, which is central to the economic and environmental viability of biosorption. This limitation creates uncertainty regarding, 1) structural degradation of the biosorbent after repeated acid or base desorption; 2) loss of functional groups, especially carboxyl and hydroxyl groups that are sensitive to hard eluents; 3) metal leaching risks, particularly for Cr⁶⁺, which poses severe toxicity concerns if released during regeneration; and 4) cost-benefit trade-offs, since regeneration efficiency often declines sharply after a few cycles. As such for future work, there is a need for systematic biosorbent regeneration studies using environmentally benign eluent and assessing structural integrity via FTIR and SEM-EDS analysis after each cycle. Future research should also explore closed-loop regeneration systems that recover metals in concentrated from for potential reuse, aligning with circular-economy principles.

5 Conclusion

The kinetic and isotherm results collectively demonstrate that BP biosorbent is a robust and versatile biosorbent capable of effectively removing both Cr⁶⁺ and Cu²⁺, albeit through distinct biosorption pathways. Cr⁶⁺ uptake was best captured by the fractal-like PFO and PSO models, with high adj. R² values (>0.994) and low χ 2 values (>1.00), indicating a mixed chemisorption-diffusion mechanism and relatively moderate capacities (≈ 2.7–3.2 mg/g). In contrast, Cu²⁺ exhibited substantially higher capacities across all models, with the F-PSO model predicting equilibrium capacities of 14.85 mg/g, reflecting strong complexation and ion-exchange interactions with the biosorbent’s heterogeneous functional groups. These kinetic trends were cemented by the investigated biosorption isotherms. Cr⁶⁺ biosorption aligned most strongly with the Langmuir-Freundlich model, suggesting heterogeneous monolayer uptake with a high theoretical capacity of Q_LF = 37.79 mg/g, while Cu²⁺ showed strong Langmuir behaviour (i.e., Q_L = 13.51 mg/g and K_L = 0.6128 L/mg), indicating high affinity and dominant monolayer chemisorption. Both the biosorption kinetics and isotherms studies confirm that BP possess a diverse array of active sites capable of binding both anionic and cationic species, with markedly higher affinity and capacity for Cu²⁺. The strong statistical conformity across kinetic and isotherm models (i.e., R² consistently >0.95 for Cr⁶⁺ and 0.90 for Cu²⁺) indicate the reliability of the biosorption process and the potential of BP biosorbent as a low-cost, sustainable material for heavy-metal remediation. By demonstrating competitive biosorption capacities, favourable kinetic behaviour, and mechanistic versatility, it can be inferred that the present study positions BP as a promising candidate for scalable wastewater treatment applications, particularly in resource-constrained setting where affordable and biodegradable adsorbents are essential.

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Footnotes

Edited by: Asif Shahzad, University of South Australia, Australia

Reviewed by: G.Vijayaraghavan GOPAL, Rajalakshmi Engineering College, India

Devu Bindhu, Amrita Vishwa Vidyapeetham (Amritapuri Campus), India