Linear and nonlinear regression methods for isotherm and kinetic modelling of iron ions bioadsorption using Ocimum sanctum Linn. leaves from aqueous solution

Since heavy metals are among the most harmful pollutants to the environment, researchers have long been interested in their presence in water and their removal (Ostad-Ali-Askari 2022). Heavy metal-laden toxic wastewater is being discharged directly or indirectly into water bodies, particularly in evolving countries, and poses a hazard to the environment because of anthropogenic activities, urbanization, and rapid industrialized development, such as metal plating services, mining/ore processing and extraction, fertiliser industries, tanneries, wood processing industries, battery manufacturing, paper and pulp industries, and pesticides (Celik & Demirbaş 2005; Costa et al. 2021; Gunarathne et al. 2022; Singh et al. 2022). Fluoride, nitrate, salinity, arsenic, iron, and other heavy metals have been reported to exceed the Bureau of Indian Standards (BIS) permissible limits in some areas of the country. Groundwater contamination by iron was reported from 301 districts across 26 states (Press Information Bureau, Government of India 2019). The Central Water Commission (CWC) conducted a study from May 2014 to April 2018 and reported that iron and copper exceeded tolerance limits in 20 river basins of the Ganga, Yamuna, and Godavari, along with other metals such as nickel, cadmium, chromium, lead, and so on. Iron and other metal contamination cases are reported from various states of Rajasthan, Uttar Pradesh, Punjab, and Tamil Nadu. One of the utmost serious environmental issues is heavy metal-induced aquatic contamination. It has extended relatively more importance, because of its toxicity, persistence, and bio-magnification.

The existence of heavy metals in the ecosystem endangers social well-being and harms organisms that live in it as well as those that consume it (Brennecke et al. 2016; Indracanti & Gunturu 2019). These heavy metals are said to be hazardous, causing harm to ecosystems as well as human health, including hypothermia, metabolic acidosis, stomach upset and ulcer, liver and brain damage, respiratory problems, mental retardation, tissue damage, and depression. Metal toxicity creates not only health problems but also an adverse effect on materials causing staining of laundry and plumbing and slimy coating on the piping because of undesirable iron bacteria growth (Sheeka Subramani et al. 2019). There are several techniques for the extraction or separation of iron from water and wastewater, namely, reverse-osmosis technique, ion exchange, chemical precipitation, adsorption technique, and membrane filtration technique (Fu & Wang 2011; Xu et al. 2021). Many of these methods were no longer sustainable for removal processes. Bioadsorption can be distinguished as the elimination of components from the arrangement by natural or biological materials (such materials can be inorganic and organic and in gaseous/vaporous form, insoluble or soluble/solvent structure). Isotherm adsorption studies were used to predict the ability of particular adsorbents to remove contaminants in the affluent approach. The lack of modern water treatment systems and a lack of skilled workers lead to water contamination and a lack of drinkable water. This study was prompted by the need for low-cost options for water and wastewater treatment. Because traditional or advanced pollution removal is impossible, low-cost water treatment options are needed for affected areas in several developing countries with poor economic conditions. Nowadays, bioadsorption is an emerging technology as it utilizes the inactive waste biomass (plant-origin biomass such as leaves, steam, seeds, shells, and agricultural waste) as an adsorbent in the adsorption process and provides maximum efficiency even at lower concentrations (Anselmo et al. 2022). The use of low-cost or economical products has become necessary in latest years due to the requirement for safe and economical approaches (De Gisi et al. 2016; Kim & Singh 2022). The requirement for safe and cost-efficient methods has necessitated the use of low-cost plant -origin biomass like Calabrian pine bark (Acemioǧlu 2004), tea leaves (Ahluwalia & Goyal 2005), Typha domingensis leaves (Abdel-Ghani et al. 2009), green clover leaves (Mem 2017), olive leaves (Elsherif 2017), Moringa oleifera seed (Abbas 2018), banana peel (Shrestha 2018a), pineapple peel (Izzati et al. 2018), and Launea procumbens leaves (Indracanti & Gunturu 2019), and agricultural by-products such as Cajanus cajan husk (Ahalya et al. 2007), pecan nutshell (Vaghetti et al. 2014), saw dust (Shrestha 2018b), hazelnut hull (Sheibani et al. 2012), and peanut hull (Awad 2017) in recent years. These are natural assets with a huge bioadsorption capacity for heavy metal expulsion from water or wastewater (Sharma et al. 2021).

Plants are considered to be an important source of medicine, and many drugs in use are derived from them, e.g. Ocimum sanctum Linn. (OSL) commonly known as Tulsi. Traditionally, O. sanctum is consumed as fresh leaves, dried powder, and herbal tea. O. sanctum is said to have a number of medicinal properties. It is a general vitalizer that improves physical endurance. Several recent studies on these extracts have revealed that they have anti-stress and anti-inflammatory, antioxidant, and immunemodulatory properties. In ayurvedic medicines, its extracts are used to treat headache, inflammation, common colds, stomach disorders, heart disease, malaria, and various forms of poisoning. It has also been proven to have anti-carcinogenic and radioprotective properties (Sreelakshmi 2017; Anitha 2018).

The current study directs to validate the viability of using OSL leaf powder as a bioadsorbent for iron ion elimination from aqueous solutions. The uptake of iron ions by OSL leaf powder was investigated in this study under various initial conditions (pH, bioadsorbent dose, bioadsorbent time, and metal concentration). To interrelate the experimental observations and to calculate the parameters of kinetics models such as pseudo-first-order model (PS1) and pseudo-second-order model (PS2), power function equation or model and Elovich model were applied. Two-parameter isotherm models such as Langmuir, Freundlich, and Temkin; three-parameter isotherm models such as Redlich–Peterson, Sips, and Toth; four-parameter models such as Baudu and Fritz–Schlunder; and five-parameter model like Fritz–Schlunder were employed to study the bioadsorption equilibrium statistics.

Linear regression is the most commonly adopted method to decide the best-fitting model for adsorption isotherm and kinetics along with the respective model parameters. However, the error distribution shifts to either the worst or the best depending on how the isotherm as well as kinetic equations are linearized. Therefore, many researchers employ the nonlinear regression approach to establish the parameters of bioadsorption isotherm and kinetic. The reduction of dispensing errors between the experimental observations and the forecasted isotherm serves as its foundation. The evaluation of isotherm data makes use of a number of error functions like the sum of square of errors (ERRSQ), the sum of absolute errors (EABS), the average relative error (ARE), the mean square error (MSE), the root-mean-square error (RMSE), and coefficient of determination R².

The thermodynamic parameters were analysed to study the adsorption process. Hence, the present work focuses on removal of iron using OSL leaf powder as bioadsorbents.

The stock iron solution was prepared by slowly adding 20 ml H₂SO₄ to 50 ml of distilled water (or di-ionized water) and then dissolving the 1.404 g of ferrous ammonium sulphate (Fe (NH₄)₂ (SO₄)₂.6H₂O). After that, 0.1 M potassium permanganate (KMnO₄) was added dropwise until a faint pink solution is formed, and then it was diluted to 1,000 ml using deionized or distilled water (APHA 2017). By diluting the stock iron metal ion solution, the essential initial concentration solution was prepared.

Bioadsorbent materials, i.e. OSL leaves in natural form, were washed by using distilled water (three to four times) to eliminate dust or dirt and any other surface contaminations. They were dried at room temperature in a clean and neat space. After that, they were dried in an oven at 60–80 °C (Shah et al. 2017). Dried adsorbents were ground and powdered using mixer and sieved through 180 μm sieve and stored in a container (labelled as OSL leaf powder) for further experimental use.

To determine physiochemical characteristics of the OSL, biomass analytical techniques were used. The morphology of OSL leaves biomass was observed using a scanning electronic microscope equipped with an energy-dispersive spectroscopy (SEM-EDS) analyser on a ZEISS-Sigma IV instrument. The Rigaku miniflex-600 instrument (having Cu Kα radiation at 40 kV and 30 mA over the range (2θ) of 20–80°, with a scanning rate at 2° per minute) was utilized to obtain X-ray diffraction (XRD) patterns. Fourier-transform infrared spectroscopy (FT-IR) bands of raw and metal-laden biomass were recorded to identify the functionalities being capable of interacting with iron ions in biomass using 400–4,000/cm spectrum on a Shimadzu IR affinity with attenuated total reflectance (ATR) having a resolution of 4/cm.

To observe the impact of pH (Labindia PICO + model used to measure pH) on the uptake percentage of iron through media by OSL leaf powder, aliquots (100 ml) comprising 0.01 g/l of the iron metal ion were shifted to a set of 250 ml conical flasks having 0.5 g of the OSL leaf powder adsorbent separately. By using 0.1 N HCl and 0.1 N NaOH solutions, the pH of each flask was adjusted between 2 and 8 and stirred at 150 rpm for 120 min at 25 °C. The biomass was disassociated from the solution by filtration, and the resulting solutions were analysed for iron by using UV-VIS spectrophotometer (Shimadzu UV1900 model was used) as mentioned in APHA (2017).

The influence of adsorbent dose, i.e. OSL leaf powder, was calculated, with adsorbent doses varying between 0.1 and 0.6 g/l. The samples were agitated for 120 min at 150 rpm at 25 °C temperature considering predefined pH, and residual concentration was analysed as mentioned earlier.

For contact time ranging between 15 and 180 min, the consequence of variation in contact time on iron ion uptake was observed with the predefined pH and bioadsorbent dose (OSL leaf powder) at 150 rpm at 25 °C temperature. The residual concentration was analysed as mentioned earlier.

The impact of the initial concentration of the solution of iron metal ions was examined by varying the concentration of the iron ion solution from 0.005 to 0.05 g/l. Predefined iron ion solutions containing varying concentrations of iron metal ions were diluted from the stock of 0.2 g/l iron standard solution (APHA 2017). The mixtures were stirred at predefined pH for 120 min at 150 rpm at 25 °C, and residual concentration was analysed as mentioned earlier.

The impact of the temperature on the uptake of iron ions was observed at different temperature ranges varying from 25 to 50 °C. The 10 mg/l initial concentration iron ion sample was agitated at predefined pH, adsorbent dose, and contact time at 150 rpm, and residual concentrations were analysed as mentioned earlier.

The time of contact required to attain equilibrium was determined through kinetic experiments. A total of 10 mg/l initial iron metal ion solutions (100 ml volume) were mixed with 0.2 g of OSL leaf powder adsorbent. Sorption experiments were conducted (at constant temperature) in flasks placed in an orbital shaker at 150 rpm till equilibria were attained. Metal concentrations were determined by collecting samples at predetermined intervals. (Unless otherwise mentioned, all experimentations were performed in triplicate. The average of results was taken from replicated tests conducted at the same temperature.)

A variety of models are available in the literature to describe sorption equilibrium. The different models are used in this study: two-, three-, four-, and five-parameter models. The isotherm model's equations and assumptions are listed in Table 1.

Table 1

Isotherm models and their assumptions

Sr. No. .	Model .	Equation .	Remarks/assumptions .
1	Two-parameter model
	Langmuir isotherm	qmax = Capacity of maximum adsorption Ce = Residual concentration at equilibrium b = constant	• Monolayer adsorption over a surface with a finite number of like sets or sites. • Adsorption energy is uniform (Langmuir 1917). • Constant signifies the affection between metal ions and bioadsorbent.
	Freundlich isotherm	K = constant for bioadsorption equilibrium n = constant (revealing of bioadsorption intensity)	• It takes surface heterogeneity and multi – layer bioadsorption to the binding sites on the sorbent's surface into account. • This model does not take biosorbent saturation into account (Freundlich 1906).
	Temkin isotherm		• The isotherm model is used for the characterization of the chemisorption system. • It is applied only for a moderate concentration range of metal ions (Shahbeig et al. 2013).
2	Three-parameter model
	Redlich–Peterson isotherm	kRP, aRP, and b = R–P parameters	• At low metal concentrations, it takes into account Henry's law, and at high concentrations, it resembles the Freundlich isotherm model (Krishna Prasad & Srivastava 2009).
	Sips isotherm	KS = Constant for sips isotherm (l/g) aS = Constant for sips model (l/mg) βS = exponent of sips isotherm	• It combines the Langmuir and Freundlich models. • At low adsorbate concentrations, it follows the Freundlich isotherm model, whereas at higher adsorbate concentrations, it obeys the Langmuir isotherm model (Saadi et al. 2015).
	Toth isotherm	KT = Toth constant for equilibrium aT = exponent for Toth isotherm	• It can be employed for the modelling of numerous heterogeneous and multilayer adsorption systems. • It is commonly used to describe various gas adsorption processes and the adsorption of organic compound vapour (Staudt 2005). Furthermore, the Toth isotherm model equation can describe data behaviour at high and low concentrations.
3	Four-parameter model
	Fritz–Schlunder isotherm	K1 and K2 = F-S model constants α and β = F-S model equilibrium constants	• It is a Langmuir–Freundlich type equation that allows for a broad range of experimental observations to be incorporated (the isotherm contains a large number of coefficients) (Ayawei et al. 2017).
	Baudu isotherm	K = Baudu model's equilibrium constant x and y = Baudu model's parameters	• It was discovered that evaluating the Langmuir coefficients (b and qm) at various equilibrium concentrations revealed that these coefficients do not remain constant over a wide range of concentrations. As a result, the Langmuir isotherm was reduced to the Baudu isotherm. • The range of (1 + x + y) < 1 and (1 + x) < 1 is covered by this isotherm model. The Freundlich isotherm model replaces this model when surface coverage is low (Nemr et al. 2010).
4	Five-parameter model
	Fritz–Schlunder isotherm	K1 and K2 = F-S model constants m and n = F-S model equilibrium constants	• It is an empirical model that can accurately simulate model variations and be used with a wide range of equilibrium statictics/data (Ayawei et al. 2017). When α and β = 1, Compared to the Freundlich isotherm model at higher adsorbate concentrations, this model is comparable to the Langmuir isotherm.

Sr. No. .	Model .	Equation .	Remarks/assumptions .
1	Two-parameter model
	Langmuir isotherm	qmax = Capacity of maximum adsorption Ce = Residual concentration at equilibrium b = constant	• Monolayer adsorption over a surface with a finite number of like sets or sites. • Adsorption energy is uniform (Langmuir 1917). • Constant signifies the affection between metal ions and bioadsorbent.
	Freundlich isotherm	K = constant for bioadsorption equilibrium n = constant (revealing of bioadsorption intensity)	• It takes surface heterogeneity and multi – layer bioadsorption to the binding sites on the sorbent's surface into account. • This model does not take biosorbent saturation into account (Freundlich 1906).
	Temkin isotherm		• The isotherm model is used for the characterization of the chemisorption system. • It is applied only for a moderate concentration range of metal ions (Shahbeig et al. 2013).
2	Three-parameter model
	Redlich–Peterson isotherm	kRP, aRP, and b = R–P parameters	• At low metal concentrations, it takes into account Henry's law, and at high concentrations, it resembles the Freundlich isotherm model (Krishna Prasad & Srivastava 2009).
	Sips isotherm	KS = Constant for sips isotherm (l/g) aS = Constant for sips model (l/mg) βS = exponent of sips isotherm	• It combines the Langmuir and Freundlich models. • At low adsorbate concentrations, it follows the Freundlich isotherm model, whereas at higher adsorbate concentrations, it obeys the Langmuir isotherm model (Saadi et al. 2015).
	Toth isotherm	KT = Toth constant for equilibrium aT = exponent for Toth isotherm	• It can be employed for the modelling of numerous heterogeneous and multilayer adsorption systems. • It is commonly used to describe various gas adsorption processes and the adsorption of organic compound vapour (Staudt 2005). Furthermore, the Toth isotherm model equation can describe data behaviour at high and low concentrations.
3	Four-parameter model
	Fritz–Schlunder isotherm	K1 and K2 = F-S model constants α and β = F-S model equilibrium constants	• It is a Langmuir–Freundlich type equation that allows for a broad range of experimental observations to be incorporated (the isotherm contains a large number of coefficients) (Ayawei et al. 2017).
	Baudu isotherm	K = Baudu model's equilibrium constant x and y = Baudu model's parameters	• It was discovered that evaluating the Langmuir coefficients (b and qm) at various equilibrium concentrations revealed that these coefficients do not remain constant over a wide range of concentrations. As a result, the Langmuir isotherm was reduced to the Baudu isotherm. • The range of (1 + x + y) < 1 and (1 + x) < 1 is covered by this isotherm model. The Freundlich isotherm model replaces this model when surface coverage is low (Nemr et al. 2010).
4	Five-parameter model
	Fritz–Schlunder isotherm	K1 and K2 = F-S model constants m and n = F-S model equilibrium constants	• It is an empirical model that can accurately simulate model variations and be used with a wide range of equilibrium statictics/data (Ayawei et al. 2017). When α and β = 1, Compared to the Freundlich isotherm model at higher adsorbate concentrations, this model is comparable to the Langmuir isotherm.

Note: q_e = capacity of adsorption (measured in mg/g).

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A variety of models are mentioned in the literature to describe kinetics. In this work, PS1, PS2, power function equation or model, and Elovich model were utilized to correlate statistics. Table 2 summarises the equations and assumptions of the kinetics models.

Table 2

Kinetics models and their assumptions

Sr. No. .	Model .	Equation .	Remarks/Assumptions .
1	Pseudo-first order	K1 = Rate constant for pseudo-first order (/min)	• The adsorption rate is corresponding to the number of empty sites by the solutes. • It only appears to work well in areas where the bioadsorption process takes place quickly (Ho & Chiang 2001).
2	Pseudo-second order	K2 = Rate constant for pseudo-second order (/min)	• Ho & McKay (1999) discovered that using the Lagergren model to estimate biosorption kinetics is not appropriate for the complete adsorption period.
3	Power function equation	k and v = adjustment parameters (mg/g/min)	• It is a quantitative representation of the relationship between sorbate mass/unit adsorbent mass of adsorbent and contact time.
4	Elovich's equation	a and b = constants	• The rate of adsorption reduces exponentially as the quantity of adsorbed solute increases. • The graphical aids in identification of the type of bioadsorption occurring on the heterogeneous superficies of the adsorbent, either chemisorption or not (Wu et al. 2009).

Sr. No. .	Model .	Equation .	Remarks/Assumptions .
1	Pseudo-first order	K1 = Rate constant for pseudo-first order (/min)	• The adsorption rate is corresponding to the number of empty sites by the solutes. • It only appears to work well in areas where the bioadsorption process takes place quickly (Ho & Chiang 2001).
2	Pseudo-second order	K2 = Rate constant for pseudo-second order (/min)	• Ho & McKay (1999) discovered that using the Lagergren model to estimate biosorption kinetics is not appropriate for the complete adsorption period.
3	Power function equation	k and v = adjustment parameters (mg/g/min)	• It is a quantitative representation of the relationship between sorbate mass/unit adsorbent mass of adsorbent and contact time.
4	Elovich's equation	a and b = constants	• The rate of adsorption reduces exponentially as the quantity of adsorbed solute increases. • The graphical aids in identification of the type of bioadsorption occurring on the heterogeneous superficies of the adsorbent, either chemisorption or not (Wu et al. 2009).

Note: q_t = capacity of adsorption at time t (mg/g); q_e = capacity of adsorption at equilibrium (mg/g).

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The OriginPro 2022 Software (version 9.9, OriginLab Corporation, MA) was utilized for the kinetic and isotherm models' linear and nonlinear regression analyses. ERRSQ, EABS, ARE, MSE, RMSE, and determination coefficient R² were implemented to examine the adequacy of each isotherm and kinetics model quantitatively (using Excel), in the present study. Table 3 summarises the equations of error functions.

Table 3

Error functions and their mathematical expressions

Sr. No. .	Error function .	Mathematical expression/formula .	References .
1	ERRSQ		Foo & Hameed (2010)
2	EABS		Nebaghe et al. (2016)
3	ARE		Nemr et al. (2010)
4	MSE		Nemr et al. (2010)
5	RMSE		Nemr et al. (2010)

Sr. No. .	Error function .	Mathematical expression/formula .	References .
1	ERRSQ		Foo & Hameed (2010)
2	EABS		Nebaghe et al. (2016)
3	ARE		Nemr et al. (2010)
4	MSE		Nemr et al. (2010)
5	RMSE		Nemr et al. (2010)

Note: q_em = experimentally measured values of bioadsorption capacities (mg/g), q_ec = computed values of bioadsorption capacities (mg/g), n = no. of data points.

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The experimental data were utilized to calculate the values of Gibbs free energy (ΔG) for various temperatures. The values of enthalpy change (ΔH) and entropy change (ΔS) were derived using slope and intercept of the graph of ln K versus 1/T.

EDS results of OSL leaf powder revealed the presence of C (53.20%), O (41.50%), Ca (1.85%), Ga (1.57%), Zn (1.06%), and Fe (0.83%) (as shown in Figure 1(c)). In the EDS spectrum of OSL leaf powder after bioadsorption (Figure 1(d)), two new peaks of iron were found to have appeared. This demonstrates that iron ions are present on the surface of OSL leaf powder. In addition, it can be seen that the distribution of the brighter areas is not uniform, signifying that only a limited number of functional groups were taking part in the bioadsorption of iron ions. It was determined from the SEM and EDS results that OSL leaf powder can absorb iron ions.

Table 4 shows the peaks extracted from FT-IR spectra. C-H, O-H, C-O, C = O, and C ≡ O functional groups were likely to be participating in the bioadsorption process of iron ions on OSL leaf powder. Along with the said group when OSL leaf powder biomass loaded with iron ions, a significant change in peaks was not observed. On the basis of FT-IR spectra, we can conclude that the chemical nature of the OSL leaf powder bioadsorbent remains almost the same after iron adsorption (Nair et al. 2013; Mandal & Bhattacharya 2015; Sakuntala et al. 2019).

The number of biosorption sites available is determined by the amount of adsorbent used. Figure 4(c) depicts the effectiveness of biosorbent concentration on metal elimination efficacy and adsorption ability. The bioadsorption of iron ions was perceived to increase (with an increase in the biosorbent quantity, an increase in biomass of surface area was observed along with the number of potential binding sites at higher doses) linearly with increasing bioadsorbent concentration up to 0.2 g (removal rate is 97.03%). After this dosage, the removal efficiency decreases. The adsorption capability (mg/g) of the OSL leaf powder decreases from 89.13 to 8.31 as a dose of bioadsorbent increases from 0.1 to 0.6 g, which may be due to the unsaturation of the active bioadsorbent sites during the processes (experimental conditions: pH = 5, initial concentration = 10 mg/l, contact time = 2 h, temperature = 25 °C, agitation speed = 150 rpm).

The usual bioadsorption kinetics show a quick initial metal uptake followed by a slower process. The extraction of iron ions increases with an increase in contact time. Figure 4(b) shows that up to 30 min of contact, there was a rapid extraction of iron ions (65.15%). Finally, the extraction decreases after 120 min. Thus, 120 min contact time was used for further studies. The maximum removal of 98.91% was observed within the first 120 min for OSL leaf powder biomass (Figure 4(b)). As contact time increases, the bioadsorption capability (in mg/g) was observed to increase (from 21.70 to 42.48). The experimental conditions applied were as follows: pH = 5, initial concentration = 0.01 g/l, bioadsorbent dose = 0.2 g/l, temperature = 25 °C, speed of agitation = 150 rpm.

The removal efficacy and uptake of iron ions of OSL leaf powder were influenced by temperature (Figure 4(e)). Both the removal efficiency and adsorption capability decrease with an increase in temperature. The separation efficiency of OSL leaf powder decreases from 98.76 to 88.23% with an increase in temperature from 25 to 50 °C. The adsorption efficiency (mg/g) decreases up to 44.11 as the temperature increases to 50 °C.

The experimental results for bioadsorption kinetics of iron ions on the OSL leaf powder were linked to kinetic models, permitting the parameters of these models to be determined as shown in Table 5 (linear and nonlinear). Correlation coefficients and error functions agreed to evaluate the model's co-relation with the experimental data. The power function equation or model and Elovich model fail to adequately describe the experimental statistics; the other two models perform better, for defining the iron ion adsorption on OSL leaf powder in both linear and nonlinear forms.

The PS1's R² values were observed to be high, i.e. (>0.99), in the case of linear analysis, while they were low for the PS2, power function model/equation, and Elovich model. More than the other mentioned kinetic models, the ERRSQ values confirm the close fitting of the experimental observations with the PS1. Consequently, linear regression was observed to be the most appropriate method for defining the kinetics of iron adsorption into OSL leaf powder in the PS1.

The nonlinear forms of all said kinetic models showed higher R² values than the linear forms. In addition, the values of the ERRSQ values were lower compared to the linear forms, across the entire range of experimental observations. The linear and nonlinear kinetic parameters of the PS1 and the PS2 were observed to vary slightly (Table 5). However, when the nonlinear equations' error function values were examined, it was discovered that they were lower than the linear form's error function values. The comparability of experimentation results with model statistics discovered a convincing correlation between the experimental observations and the PS1, which was confirmed to be the superlative for explaining bioadsorption kinetics of iron ions on the OSL leaf powder.

The inter-relationship among the quantity of the bioadsorbed constituent per bioadsorbent quantity and the concentration of this constituent in the solution is represented by bioadsorption isotherms. The estimation of equilibrium parameters affords important facts and statistics that can be used to design future adsorption structures or systems.

The experimental statistics was compared with 3 two-parameter, 3 three-parameter, 2 four-parameter, and 1 five-parameter models that described the bioadsorption equilibrium of iron ions on OSL leaf powder are given in Table 6 (linear) and Table 7 (nonlinear).

Table 6 shows a comparison of the results of the isotherm by the linear method. The close-fitted isotherm is chosen using error functions based on the error functions that resulted in a lower error between the experimental and calculated amounts of iron bioadsorbed and a higher correlation coefficient. When the error equations (ERRSQ, EABS, ARE, MSE, RMSE, and R²) were taken into consideration, the Langmuir isotherm model was found to have the lowest error values. In the linear method, the Redlich–Peterson isotherm, Sips and Toth isotherm, Freundlich isotherm, and Temkin isotherm were followed by those models.

For the nonlinear method, a computer-operable trial-and-error method was utilized to minimize the error dispensation between the experimental observations and the isotherms studied. The error functions that correspond to the parameters of the calculated isotherm are shown in Table 7. It demonstrates that the Langmuir isotherm, followed by the Freundlich–Temkin isotherm (for two-parameter models), was identified as the close-fit isotherm by the error functions that were correlated to the smallest deviations from the predicted equilibrium data (Figure 7). It could be found that the Langmuir model very well represents the experimental observations on the bioadsorption of iron ions onto OSL leaf powder. The Langmuir model has the lowest error values and relatively high R². The q_max value (in mg/g) for iron ions was 123.26. In the Freundlich equilibrium, the k constants for OSL leaf powder were obtained to be 73.75 l/g. The value of n obtained is between 0 and 10, which signifies that iron ions adsorb relatively strongly on the superficies of the adsorbents. The low correlation coefficients, on the other hand, showed that this was a unsuitable model for describing this equilibrium. It can be seen that the error function values produced by the nonlinear method are lower when compared to those produced by the linear method. The error functions for the nonlinear regression method have not changed except for the Temkin isotherm model. The great similarity between Temkin model's linear and nonlinear equation forms may account for the same.

The equilibrium adsorption statics data can be modelled by the three-parameter isotherm equations, i.e. Redlich–Peterson, Sips, and Toth isotherms. The isotherm parameters derived from the linear and nonlinear fitting analyses are summarized in Tables 6 and 7. The coefficients of determination are very good (R² > 0.990) for all the three models (Table 7). R² values for the Redlich–Peterson isotherm model were observed to be 0.99 for both nonlinear and linear model fitting but show lower error values in the case of the nonlinear models. A superior and comprehensive representation of the experimental data for the bioadsorption isotherms was observed for the Sips isotherm model (considering values of the error functions). The Sips and Toth model's linear and nonlinear trends show a poor fit for the linear model and a strong correlation for the nonlinear model (by comparing the values of coefficient of determination, i.e. R² and other error functions). The exponent of the Toth model (t) calculated by linear and nonlinear regression falls in the range of 0–1, suggesting a good resemblance of Toth isotherm with Langmuir isotherm. Adsorption occurred on a heterogeneous surface when t is less than 1. The close-fitted bioadsorption isotherm models were found to be in the following order observed using the three-parameter equations: Sips > Toth = Redlich–Peterson. The equilibrium statistics or data were well described by three-parameter isotherm models than by two-parameter isotherm models, as shown earlier.

With linear and nonlinear four-parameter isotherm models, the adsorption equilibrium statistics were examined. The four-parameter isotherm model of the Fritz–Schlunder isotherm and the Baudu isotherm is used to obtain an accurate fit to the experimental results of bioadsorption isotherms (Figure 9). Fritz–Schlunder isotherm shows that the coefficients of determination (R² > 0.99) are very good and have low values for error function as compared to Baudu (R² = 0.86). The adsorption of iron ions on OSL leaf powder appears to be better explained by Fritz–Schlunder's equation than the Baudu isotherm. The Baudu isotherm has a lower maximum bioadsorption capacity than the Langmuir isotherm, while it is greater than the Langmuir isotherm and theoretical values in the case of the Fritz–Schlunder isotherm.

The nonlinear form of the Fritz–Schlunder five-parameter isotherm model was used to analyse the adsorption data. The Fritz–Schlunder five-parameter isotherm model provides a satisfactory fit to the bioadsorption isotherms' experimental observations (Figure 10). The Fritz–Schlunder five-parameter models fit better than the Baudu isotherm model. (The correlation coefficients are very high (>0.99), and the error functions have low values).

Table 8 compares the biosorption uptake of iron ions by OSL leaf powder and other biosorbents mentioned in the literature. The adsorbent shows the potential for the separation of iron ions from the aqueous media.

Because of differences in surface properties interconnected to the existence of several functional groups, the sorption abilities of specific biological materials may differ.

The negative values of change in Gibb's free energy (ΔG) found for the bioadsorption of iron ions by OSL leaf powder at various temperatures firm up the feasibility of the reaction and revealed the spontaneous kind of the bioadsorption processes, which did not involve an external energy source for the system. The negative value of ΔH (difference in enthalpy) indicates adsorption reaction to be in agreement with the exothermic nature of interactions, and the negative value of ΔS (change in entropy) presented the decreased uncertainty at solid–liquid interphase throughout the bioadsorption processes of iron ions on the said adsorbent. Negative values of ΔH and ΔS revealed that no structural changes of the adsorbent, which supports adsorption of iron ions on the OSL leaves powder was physical in nature.

OSL leaf powder is an environmental friendly and cost-effective biosorbent for heavy metals removal from aqueous stream or media. This study examines the effectiveness of OSL leaf powder for the extraction of iron ions from aqueous media.

The kinetics and equilibrium of iron ions bioadsorption onto the surface of OSL leaf powder were described using several mathematical models. The maximum potential of bioadsorption (in mg/g) of 123.26 was calculated from the Langmuir model for OSL leaf powder, under optimum conditions (pH 5, biomass concentration 0.2 g, contact time 2 h and speed of agitation 150 rpm at temperature 25 °C). The possible, spontaneous, and exothermic nature of the bioadsorption process of iron metal ions on OSL leaf powder was revealed by the thermodynamic parameters. The OSL leaf powder has been effectual in extracting iron ions from aqueous media.

The outcome of this study shows that, in comparison to linear regression, nonlinear regression is the most effective method for finding the ideal isotherm and the parameters of the isotherm. The Fritz–Schlunder isotherm, derived from a comparison of nonlinear isotherms, is the close-fitting model for the bioadsorption of iron ions on powdered OSL leaves. For the purpose of describing the adsorption equilibrium isotherms of iron ions on powdered OSL leaves, all of the tested models were categorized as follows: Fritz–Schlunder (five parameters) = Fritz–Schlunder (four parameters) > Sips > Redlich–Peterson > Toth = Langmuir > Temkin > Baudu = Freundlich. According to kinetic experiments, it was observed that the bioadsorption process is quick and attained equilibrium in a short contact time. The PS1 model (nonlinear) well defined the process's kinetics of bioadsorption of iron ions on OSL leaves adsorbents. When nonlinear kinetic equations are transformed into linear forms, the parameters of the model may be distorted. Furthermore, the nonlinear kinetic equations have the advantage of not necessitating prior knowledge of q_e (experimental) to fit the experimental points. Thus, the adsorption parameters can be primarily obtained using the nonlinear approach. In addition, the ERRSQ approach of error analysis may be superior to use for the determination of correlation coefficient (R²) and to identify close-fitting models. Upscaling and commercialization of the adsorbent on a pilot-scale can be studied, as well as its regeneration, reuse, and secure disposal of the laden adsorbent can be addressed in future studies. Furthermore, research on the bioadsorption of iron may concentrate on the topics like desorption of bioadsorbent, life cycle assessment of bioadsorbent, analysing the cost associated with a potential large-scale system for industrial application, and iron bioadsorption using bionanomaterials from alternate sources.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.