Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae: Adsorption Modelling and Applicability of Various Isotherm Models

Abstract

The adsorption of Ni(II) onto blue green marine algae (BGMA) in batch conditions is being investigated. The highest adsorption capacity of BGMA was found to be 42.056 mg/g under ideal testing conditions, where the initial Ni(II) metal ion concentration was adjusted from 25 ppm to 250 ppm. The optimal pH, biomass loading, and agitation rate for maximum Cu(II) ion removal have been determined to be 6, 2 g and 120 rpm, respectively. For the equilibrium condition, 24 hours of contact time is allowed. At room temperature, all of the experiments are conducted. The isotherm has a L shape, based on the equilibrium experimental data. It indicates that there is no considerable competition for active sites between the solvent and Ni(II). There is no strong competition between the solvent and Ni(II) for the active sites of BGMA, indicating that there is no strong competition between the two. It also suggests that the BGMA’s Ni sorption ability is restricted (II). The experimental data is validated using multiple isotherm models, and the mechanism of adsorption is then discovered, as well as the process design parameters. The Fritz-Schlunder-V isotherm model is particularly relevant in defining the mechanism of Ni(II) adsorption under the conditions used in this study, according to modelling studies. This model’s qmax of 41.89 mg/g shows that it matches experimental data more closely.

1. Introduction

Metal-processing industries would undoubtedly have a challenge in disposing of metal-bearing effluents. Most industries dishonestly discharge their effluents into surrounding drains and water streams, either untreated or partially treated [1] due to increase in overall industrial cost. Industries such as alloys, pigments, electroplating, mining, metallurgical activities, nuclear power plant operations, aerospace industries, electrical contacts, printing, and the manufacture of paper, rubber, plastics, and batteries play a major role in water pollution by releasing heavy metal ions in their effluents [2].

Nickel, a non-biodegradable hazardous metal ion, is the heavy metal ion studied in this work [3]. Ni(II) ions in drinking water are allowed to be at a concentration of 0.02 mg/L. Anaemia, diarrhoea, encephalopathy, hepatitis, lung and kidney damage, gastrointestinal distress, pulmonary fibrosis, renal edoema, skin dermatitis, and central nervous system dysfunction are just some of the negative health impacts of exceeding the permissible limit [4]. As a result, before being discharged into the environment, industrial effluents containing Ni(II) must be treated [5].

Traditional methods for removing Ni(II) metal ions include coagulation, electro dialysis, flotation, ion exchange, precipitation, reverse osmosis, and others [6]. Low competency performances, especially when using these methods on very small concentrations of metal ions [7], are some of the limitations of these traditional approaches.

Adsorption with a low-cost sorbent is a frequently used approach in the treatment of industrial effluents [8]. However, it is still necessary to develop a low-cost, readily available, high-adsorption-capacity waste water treatment material that can address the aforementioned environmental concerns [9]. Because it is efficient, avoids secondary wastes, and utilises low-cost resources, biosorption onto live or non-living biomass, such as fungi, bacteria, yeast, moss, aquatic plants, and algae, can be a viable approach for removing heavy metals from their source [10]. Marine algae in coastal locations play a significant role in world ecology, are exceedingly efficient, and are taxonomically varied [11]. Many sections of the world gather or produce marine macroalgae, making them easily accessible in huge amount in the manufacture of very efficient bio sorbent resources [12]. The goal of this study is to utilise Blue Green Marine Algae (BGMA) as an adsorbent to adsorb the metal ions Ni(II) present in an artificial aqueous solution [13].

There is a scarcity of relevant literature on Ni(II) adsorption with three, four, and five parameter models [14, 15]. The constraints of the simple, one- and two-parameter models would be overcome by the high-parameter models. The use of large parameter models to describe the adsorption process under equilibrium conditions can provide highly clear and accurate information. The purpose of this work is to determine the biosorption capacity of BGMA for the removal of Ni(II) metal ions from a synthetically generated stock solution under optimal experimental circumstances of pH 6, 2 g biomass loading, and 120 rpm agitation speed. In addition, the experimental data is examined using one, two, three, four, and five parameter isotherm models. For the purpose of modelling, the experimental data is examined using one, two, three, four, and five parameter isotherm models.

2. Materials and methods

The BGMA was collected along the coast of Tamil Nadu, India, near Chidambaram. It is cleaned and dried at room temperature with distilled water. It is then pulverised to 150–200 microns in size. For IR spectrum investigations of dried biomass and Ni(II)-sorbed biomass in the range of 4000–400 cm⁻¹, a Fourier transform infrared (FT-IR) spectrometer (BRUKER FT-IR, ALPHA-T, GERMANY) was employed.

The synthetic Ni(II) solution is made with an analytical grade salt, Nickel(II) sulfate heptahydrate (NiSO₄·7H₂O). To make 1 L of solution for the stock purpose of 1000 ppm, exactly 4.7852 g of NiSO₄·7H₂O is weighed and utilised. For stock solution preparation, double distilled water is employed. This stock solution is diluted to a concentration of 25–250 parts per million.

Each solution’s pH is changed from 2 to 7. pH is adjusted with 0.1 N nitric acid (HNO₃) and 0.1 N sodium hydroxide (NaOH) solutions. The amount of biomass put to the conical flask varies from 0.5 to 2.5 g (0.5, 1.0, 1.5, 2.0 and 2.5 g).

A 500 mL conical flask is used for the batch adsorption experiment. The starting concentrations of metal ions are 25, 50, 75, 100, 125, 150, 175, 200, 225, and 250 parts per million. Each 500 mL Erlenmeyer flask contains 400 mL of 25 ppm metal ion solution. Each Erlenmeyer flask receives 0.5, 1.0, 1.5, 2.0, and 2.5 g of BGMA, respectively. In all five flasks, the pH of the solution is kept at 2. The flasks are agitated at 120 rpm in a rotary shaker. There is a total of 24 hours of contact (shaking) time given. This is more than enough to attain the desired equilibrium (maximum adsorption). The starting and ultimate concentrations of the solution are determined using a double-beam Atomic Adsorption Spectrophotometer (AAS SL176-Elico Limited India). The % removal of metal ions is computed using Eq. (1) from the starting (C_in) and equilibrium final concentration (C_eq).

$$\%\text{Removal}=\frac{{\mathrm{C}}_{\text{in}}-{\mathrm{C}}_{\mathrm{eq}}}{{\mathrm{C}}_{\text{in}}}\times 100$$E1

Eq. (2) is used to compute the equilibrium metal uptake, q_eq, using the starting (C_i) and equilibrium (C_eq) concentrations of the metal ion solution.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{\mathrm{V}}{\mathrm{M}}\left({\mathrm{C}}_{\text{in}}-{\mathrm{C}}_{\mathrm{eq}}\right)$$E2

where V is the litre volume of the liquid sample and M is the gram weight of the adsorbent. In order to improve the pH value and biomass loading, the same operation is repeated with adjusting the solution pH as 3, 4, 5, 6 and 7. Similarly, 50 ppm, 75 ppm, 100 ppm, 125 ppm, 150 ppm, 175 ppm, 200 ppm, 225 ppm, and 250 ppm metal ion concentrations are used in the studies. For concordant results, experiments are repeated (Table 1).

S. no.	Cin (mg/L)	Ceq (mg/L)	qeq (mg/g)	Removal (%)
1	25	1.52	4.696	93.92
2	50	3.25	9.350	93.50
3	75	5.23	13.954	93.03
4	100	6.54	18.692	93.46
5	125	15.39	21.922	87.69
6	150	21.58	25.684	85.61
7	175	27.64	29.472	84.21
8	200	29.39	34.122	85.31
9	225	35.79	37.842	84.09
10	250	39.72	42.056	84.11

Table 1.

Experimental values of adsorption of Ni(II) onto BGMA.

One parameter model (Henry’s law), two parameter models (Henry’s law with intercept, Langmuir, Freundlich, Dubinin-Radushkevich, Temkin, Hill-de Boer, Fowler-Guggenheim, Flory-Huggins, Halsey, Harkin-Jura, Jovanovic, Elovich and Kiselev), three parameter models (Hill, Redlich-Peterson, Sips, Langmuir-Freundlich, Fritz-Schlunder-III, Radke-Prausnits-I, Radke-Prausnits-II, Radke-Prausnits-III, Toth, Khan, Koble-Corrigan, Jossens, Jovanovic-Freundlich, Brouers-Sotolongo, Vieth-Sladek, Unilan, Holl-Krich and Langmuir-Jovanovic), four parameter models (Fritz-Schlunder-IV, Baudu Weber-van Vliet and Marczewski-Jaroniec) and five parameter model (Fritz-Schlunder-5) are used to examine the experimental facts and to discover its applicability for modelling purpose. The parameter values are predicted using the cftool kit available in MATLAB R2010a software. This toolkit aids in the estimation of model parameters, including the non-linear regression coefficient (R²), Sum of Squares due to Error (SSE), and Root Mean Squared Error (RMSE).

3. Isotherm models: theoretical knowledge

Isotherm models are used to determine an adsorbent’s maximal sorption capacity and is represented in terms of the amount of metal absorbed per unit mass of adsorbent.

3.1 One parameter model

The simple adsorption isotherm model has only one parameter to explain the adsorption mechanism. The most basic adsorption isotherm is one in which the amount of solute adsorbed is proportional to the equilibrium effluent concentration [16]. Eq. (3) is the model.

$${\mathrm{q}}_{\mathrm{eq}}=\mathrm{K}{\mathrm{C}}_{\mathrm{eq}}$$E3

3.2 Two parameter models

This section focuses on gaining theoretical insight into models with two parameters that explain the adsorption mechanism. Henry’s law with constant, Langmuir, Freundlich, Dubinin-Radushkevich, Temkin, Hill-de Boer, Fowler-Guggenheim, Flory-Huggins, Halsey, Harkin-Jura, Jovanovic, Elovich, and Kiselev are among the 13 models covered.

3.2.1 Henry’s law with intercept isotherm model

This model was created to address the contradiction highlighted by the one parameter model and to be applicable over a wide range of metal ion concentrations [16]. Eq. (4) describes the model.

$${\mathrm{q}}_{\mathrm{eq}}=\mathrm{K}{\mathrm{C}}_{\mathrm{eq}}+\mathrm{C}$$E4

3.2.2 Langmuir isotherm model

The Langmuir model assumes a homogenous surface and explains how the adsorbate forms monolayer coverage on the adsorbent’s outer surface [17, 18]. The rate of adsorption is proportional to the percentage of open adsorbent surface, and the rate of desorption is related to the fraction of covered adsorbent surface. The model is described in Eq. (5).

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{b}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{eq}}}{1+{\mathrm{b}}_{\mathrm{L}}{\mathrm{C}}_{\mathrm{eq}}}$$E5

b_L is the Langmuir constant, which links the fluctuation of the appropriate area and porosity of the adsorbent with its adsorption capacity (mg/g). A dimensionless constant called the Langmuir separation factor R_L, which is computed as Eq. (6), helps explain the key properties of the Langmuir isotherm.

$${\mathrm{R}}_{\mathrm{L}}=\frac{1}{1+{\mathrm{b}}_{\mathrm{L}}{\mathrm{q}}_{\max }}$$E6

When R_L > 1, adsorption is unfavourable, when R_L = 1, linear when R_L = 1, favourable when R_L = 0, and irreversible when R_L = 0.

3.2.3 Freundlich isotherm model

The Freundlich adsorption isotherm model depicts the adsorbent surface heterogeneity. The adsorptive sites are made up of tiny homogenous heterogeneous adsorption sites [19]. Eq. (7) is the model.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{a}}_{\mathrm{F}}{\mathrm{C}}_{\mathrm{eq}}^{\frac{1}{{\mathrm{n}}_{\mathrm{F}}}}$$E7

Freundlich adsorption capacity (L/mg) is denoted by a_F, while adsorption intensity is denoted by n_F. The higher the adsorption capacity, the larger the a_F value. The magnitude of 1/n_F, which varies from 0 to 1, indicates favourable adsorption and becomes more heterogeneous as it approaches zero [18, 19, 20, 21].

3.2.4 Dubinin-Radushkevich isotherm model

This empirical model implies that physical adsorption processes are multilayered and involve Van Der Waal’s forces [22], it is frequently used to estimate the characteristic porosity. This model [23] gives insight into gas and vapour adsorption on micro porous sorbents.

The Dubinin-Radushkevich isotherm’s temperature dependence is another distinguishing trait [24, 25]. Eq. (8) represents the Dubinin-Radushkevich isotherm.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{K}}_{\mathrm{DR}}exp\left[-{\mathrm{B}}_{\mathrm{DR}}{\left(\text{RTln}\left(1+\frac{1}{{\mathrm{C}}_{\mathrm{eq}}}\right)\right)}^2\right]$$E8

$$\mathrm{\epsilon }=\text{RTln}\left[\frac{1}{{\mathrm{C}}_{\mathrm{eq}}}\right]$$E9

ε is known as the Polanyi potential, as seen in Eq. (9). The activation energy or mean free energy E (kJ/mol) of adsorption per molecule of adsorbate when it is transported from infinity in the solution to the surface of the solid may be computed using Eq. (10).

$$\mathrm{E}=\frac{1}{\sqrt{2{\mathrm{K}}_{\mathrm{DR}}}}$$E10

The value of E is used to forecast whether an adsorption is physical or chemical. In nature, physisorption occurs when E = 8 KJ/mol, whereas chemisorption occurs when E = 8–16 KJ/mol [23].

3.2.5 Temkin isotherm model

For forecasting the gas phase adsorption equilibrium, the Temkin adsorption isotherm model is quite useful [25, 26]. Eq. (11) illustrates the Temkin adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{\mathrm{RT}}{{\mathrm{b}}_{\mathrm{T}}}\left(ln{\mathrm{A}}_{\mathrm{T}}{\mathrm{C}}_{\mathrm{eq}}\right)$$E11

The heat of adsorption is represented by the equation RT/b_T, and the equilibrium binding constant (L/mg) corresponding to the maximal binding energy is represented by A_T.

3.2.6 Hill-de Boer isotherm model

This Hill-Deboer isotherm model accurately describes mobile adsorptions as well as lateral interactions between adsorbed molecules [27, 28]. Eq. (12) depicts the Hill-Deboer isotherm model.

$${\mathrm{K}}_1{\mathrm{C}}_{\mathrm{eq}}=\frac{\mathrm{\theta }}{1-\mathrm{\theta }}exp\left(\frac{\mathrm{\theta }}{1-\mathrm{\theta }}-\frac{{\mathrm{K}}_2\mathrm{\theta }}{\mathrm{RT}}\right)$$E12

A positive value of K₂ implies attraction between adsorbed species, whereas a negative value of K₂ suggests repulsion. If K₂ is equal to zero, there is no interaction between adsorbed molecules, and the Volmer equation [29] is used

3.2.7 Fowler-Guggenheim isotherm model

The Fowler-Guggenheim adsorption isotherm model describes how adsorbed molecules interact laterally. Eq. (13) represents the Fowler-Guggenheim adsorption isotherm.

$${\mathrm{K}}_{\mathrm{FG}}{\mathrm{C}}_{\mathrm{eq}}=\frac{\mathrm{\theta }}{1-\mathrm{\theta }}exp\left[\frac{2\mathrm{\theta W}}{\mathrm{RT}}\right]$$E13

The presence of a positive W indicates that the contact between the adsorbed molecules is attractive. In contrast, if W = 0, the contact between adsorbed molecules is repulsive, and the heat of adsorption decreases with loading, the Fowler-Guggenheim equation reduces to the Langmuir model. W = 0 when there is no contact between adsorbed molecules. Furthermore, this model is only viable when the surface coverage is less than 0.6 [30, 31].

3.2.8 Flory-Huggins isotherm model

The degree of surface coverage of the adsorbate onto the adsorbent is discussed by the Flory-Huggins isotherm [32, 33, 34, 35]. Eq. (14) shows the Flory-Huggins adsorption isotherm.

$$\frac{\mathrm{\theta }}{{\mathrm{C}}_{\text{in}}}={\mathrm{K}}_{\mathrm{FH}}{\left[1-\mathrm{\theta }\right]}^{{\mathrm{n}}_{\mathrm{FH}}}$$E14

K_FH is used to calculate the spontaneity Gibbs free energy.

3.2.9 Halsey isotherm model

In multi layer adsorption, the hetero porous nature of the adsorbent is demonstrated by the fitting of experimental data to this equation [36]. Eq. (15) denotes the Halsey adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=exp\left[\frac{ln{\mathrm{K}}_{\mathrm{Ha}}-ln{\mathrm{C}}_{\mathrm{eq}}}{{\mathrm{n}}_{\mathrm{Ha}}}\right]$$E15

3.2.10 Harkin-Jura isotherm model

The Hurkin-Jura adsorption isotherm is used to explain multilayer adsorption on the surface of absorbents with heterogeneous pore distribution [37]. Eq. (16) describes the Hurkin-Jura adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\sqrt{\frac{{\mathrm{A}}_{\mathrm{HJ}}}{{\mathrm{B}}_{\mathrm{HJ}}-log{\mathrm{C}}_{\mathrm{eq}}}}$$E16

3.2.11 Jovanovic isotherm model

With the approximation of monolayer localised adsorption without lateral contacts, the Jovanovic adsorption isotherm model is analogous to the Langmuir model [38]. Eq. (17) shows the Jovanovic adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{q}}_{\max }\left[1-{\mathrm{e}}^{-\left({\mathrm{K}}_{\mathrm{J}}{\mathrm{C}}_{\mathrm{eq}}\right)}\right]$$E17

It forms the Langmuir isotherm at large adsorbate concentrations but does not obey Henry’s rule.

3.2.12 Elovich isotherm model

Adsorption sites expand exponentially with adsorption, demonstrating multilayer adsorption, according to the Elovich isotherm model [39]. The Elovich adsorption isotherm model is depicted in Eq. (18).

$$\frac{{\mathrm{q}}_{\mathrm{eq}}}{{\mathrm{q}}_{\max }}={\mathrm{K}}_{\mathrm{E}}{\mathrm{C}}_{\mathrm{eq}}{exp}^{\frac{{\mathrm{q}}_{\mathrm{eq}}}{{\mathrm{q}}_{\max }}}$$E18

3.2.13 Kiselev isotherm model

The Kiselev adsorption isotherm model is also known as localised monomolecular layer model [40]. This model is valid only for surface coverage θ > 0.68. The Kiselev adsorption isotherm model is given in Eq. (19).

$${\mathrm{K}}_{\mathrm{eqK}}{\mathrm{C}}_{\mathrm{eq}}=\frac{\mathrm{\theta }}{\left(1-\mathrm{\theta }\right)\left(1+{\mathrm{K}}_{\mathrm{nK}}\mathrm{\theta }\right)}$$E19

3.3 Three parameter models

Models containing three parameters to explain the mechanism of adsorption are discussed using 16 models viz., Hill, Redlich-Peterson, Sips, Langmuir-Freundlich, Fritz-Schlunder-III, Radke-Prausnits, Toth, Khan, Koble-Corrigan, Jossens, Jovanovic-Freundlich, Brouers-Sotolongo, Vieth-Sladek, Unilan, Holl-Krich and Langmuir-Jovanovic.

3.3.1 Hill isotherm model

To characterise the adhesion of diverse species to homogeneous substrates, the Hill isotherm model is developed [41]. Eq. (20) depicts the Hill adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{H}}}}{{\mathrm{K}}_{\mathrm{H}}+{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{H}}}}$$E20

If n_H is greater than 1, this isotherm indicates positive co-operativity in binding, n_H is equal to 1, it indicates non-cooperative or hyperbolic binding and n_H is less than 1, indicating negative co-operativity in binding.

3.3.2 Redlich-Peterson isotherm model

The Redlich-Peterson isotherm model is created by combining elements of the Langmuir and Freundlich isotherms [42]. Eq. (21) depicts the Redlich-Peterson isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{A}}_{\mathrm{RP}}{\mathrm{C}}_{\mathrm{eq}}}{1+{\mathrm{B}}_{\mathrm{RP}}{\mathrm{C}}_{\mathrm{eq}}^{\mathrm{\beta }}}$$E21

When the liquid phase concentration is low, this model approaches Henrys Law. β_RP, the exponent, is usually between 0 and 1. When β_RP = 1, this model is similar to the Langmuir model, and when β_RP = 0, this isotherm is similar to the Freundlich model.

3.3.3 Sips isotherm model

The Sips adsorption isotherm model [43] was designed to represent localised adsorption without adsorbate-adsorbate interactions [44] at high adsorbate concentrations. Eq. (22) is the Sips adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\mathrm{m}}{\mathrm{K}}_{\mathrm{s}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\beta }}_{\mathrm{S}}}}{1+{\mathrm{K}}_{\mathrm{s}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\beta }}_{\mathrm{S}}}}$$E22

When β_S equal to 1 this isotherm approaches Langmuir isotherm and β_S equal to 0, this isotherm approaches Freundlich isotherm.

3.3.4 Langmuir-Freundlich isotherm model

Adsorption in heterogeneous surfaces is described by the Langmuir-Freundlich isotherm model [17, 18]. Eq. (23) depicts the Langmuir-Freundlich isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\left({\mathrm{K}}_{\mathrm{LF}}{\mathrm{C}}_{\mathrm{eq}}\right)}^{{\mathrm{m}}_{\mathrm{LF}}}}{1+{\left({\mathrm{K}}_{\mathrm{LF}}{\mathrm{C}}_{\mathrm{eq}}\right)}^{{\mathrm{m}}_{\mathrm{LF}}}}$$E23

m_LF is a heterogeneous parameter with a value between 0 and 1. This value rises when the degree of surface heterogeneity decreases. For m_LF is equal to 1, this model covert to Langmuir model.

3.3.5 Fritz-Schlunder-III isotherm model

Because of the large number of coefficients in their isotherm, the Fritz-Schunder three parameter isotherm model was constructed to suit a wide variety of experimental findings [45]. Eq. (24) has this expression.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{K}}_{\mathrm{FS}3}{\mathrm{C}}_{\mathrm{eq}}}{1+{\mathrm{q}}_{\max }{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{m}}_{\mathrm{FS}3}}}$$E24

If m_FS3 is equal to 1, the Fritz-Schlunder-III model becomes the Langmuir model but for high concentrations of adsorbate, the Fritz-Schlunder-III reduces to the Freundlich model.

3.3.6 Radke-Prausnits isotherm model

At low adsorbate concentrations, the Radke-Prausnits isotherm model has numerous essential qualities that make it preferable in most adsorption systems [46]. When the Radke-Prausnits model exponent mRaP3 is equal to zero. Eqs. (25)–(27) show Radke-Prausnits isotherm models.

$$\text{Model}1:{\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{K}}_{\mathrm{RaP}1}{\mathrm{C}}_{\mathrm{eq}}}{{\left[1+{\mathrm{K}}_{\mathrm{RP}1}{\mathrm{C}}_{\mathrm{eq}}\right]}^{{\mathrm{m}}_{\mathrm{RaP}1}}}$$E25

$$\text{Model}2:{\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{K}}_{\mathrm{RaP}2}{\mathrm{C}}_{\mathrm{eq}}}{1+{\mathrm{K}}_{\mathrm{RP}2}{{\mathrm{C}}_{\mathrm{eq}}}^{{\mathrm{m}}_{\mathrm{RaP}2}}}$$E26

$$\text{Model}3:{\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{K}}_{\mathrm{RaP}3}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{m}}_{\mathrm{RaP}3}}}{1+{\mathrm{K}}_{\mathrm{RP}3}{{\mathrm{C}}_{\mathrm{eq}}}^{{\mathrm{m}}_{\mathrm{RaP}3-1}}}$$E27

When both m_RaP1 and m_RaP2 are equal to 1, the Radke-Prausnitz 1, 2 models decrease to the Langmuir model; nevertheless, at low concentrations, the models become Henry’s law; but, at high adsorbate concentrations, the Radke-Prausnitz 1 and 2 models become the Freundlich model. When the exponent m_RaP3 is equal to 1, the Radke-Prausnitz-3 equation simplifies to Henry’s law, and when the exponent mRaP3 is equal to 0, it becomes the Langmuir isotherm.

3.3.7 Toth isotherm model

The Toth adsorption isotherm model is used to explain heterogeneous adsorption systems that fulfil both the low and high end boundaries of adsorbate concentration [47]. Eq. (28) represents the Toth isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{C}}_{\mathrm{eq}}}{{\left(\frac{1}{{\mathrm{K}}_{\mathrm{T}}}+{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{T}}}\right)}^{\frac{1}{{\mathrm{n}}_{\mathrm{T}}}}}$$E28

When n = 1, this equation simplifies to the Langmuir isotherm equation, indicating that the process is approaching the homogenous surface. As a result, the value n describes the adsorption system’s heterogeneity. The system is considered to be heterogeneous if it deviates farther from unity.

3.3.8 Khan isotherm model

For adsorption of bi-adsorbate from pure dilute equations solutions, the Kahn isotherm model is devised [48]. Eq. (29) presents the Kahn isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\mathrm{m}}{\mathrm{b}}_{\mathrm{K}}{\mathrm{C}}_{\mathrm{eq}}}{{\left(1+{\mathrm{b}}_{\mathrm{K}}{\mathrm{C}}_{\mathrm{eq}}\right)}^{{\mathrm{a}}_{\mathrm{K}}}}$$E29

When a_K is equal to one, the Toth model approaches the Langmuir isotherm model, and when a_K is more than one, the Toth model simplifies to the Freundlich isotherm model.

3.3.9 Koble-Corrigan isotherm model

The Sips isotherm model is similar to the Koble-Carrigan isotherm model. This model includes both the Langmuir and the Freundlich isotherms [49]. Eq. (30) depicts the Koble-Carrigan isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{A}}_{\mathrm{KC}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{KC}}}}{1+{\mathrm{B}}_{\mathrm{KC}}{{\mathrm{C}}_{\mathrm{eq}}}^{{\mathrm{n}}_{\mathrm{KC}}}}$$E30

This model reduces to the Freundlich isotherm at large adsorbate concentrations. It is only acceptable when n is higher than or equal to 1. When n is lesser than one, it indicates that the model, despite a high concentration coefficient or a low error value, is incapable of describing the experimental data.

3.3.10 Jossens isotherm model

The Jossens isotherm model is based on the energy distribution of adsorbate-adsorbent interactions at adsorption sites [50]. At low concentrations, this model is reduced to Henry’s law. Eq. (29) depicts the Jossens isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{K}}_{\mathrm{J}}{\mathrm{C}}_{\mathrm{eq}}}{1+\mathrm{J}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{b}}_{\mathrm{J}}}}$$E31

At low capacity, J equates to Henry’s constant. b_J is the Jossens isotherm constant, which is constant regardless of temperature or the composition of the adsorbent.

3.3.11 Jovanovic-Freundlich isotherm model

To depict single-component adsorption equilibrium on heterogeneous surfaces, the Jovanovic-Freundlich isotherm model is developed [51]. Eq. (32) represents the Jovanovic-Freundlich isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{q}}_{\max }\left[1-{\mathrm{e}}^{-\left({\mathrm{K}}_{\mathrm{JF}}{{\mathrm{C}}_{\mathrm{eq}}}^{{\mathrm{n}}_{\mathrm{JF}}}\right)}\right]$$E32

3.3.12 Brouers-Sotolongo isotherm model

This isotherm is built in the form of a deformed exponential function for adsorption onto a heterogeneous surface [52]. Eq. (33) depicts the Brouers-Sotolongo model.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{q}}_{\max }\left[1-{\mathrm{e}}^{\left(-{\mathrm{K}}_{\mathrm{BS}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\alpha }}_{\mathrm{BS}}}\right)}\right]$$E33

The parameter α_BS is related with distribution of adsorption energy and the energy of heterogeneity of the adsorbent surfaces at the given temperature [53].

3.3.13 Vieth-Sladek isotherm model

This model includes two independent parts for calculating transient adsorption diffusion rates in solid adsorbents [54]. Eq. (34) represents the Vieth-Sladek isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{K}}_{\mathrm{VS}}{\mathrm{C}}_{\mathrm{eq}}+\frac{{\mathrm{q}}_{\max }{\mathrm{\beta }}_{\mathrm{VS}}{\mathrm{C}}_{\mathrm{eq}}}{1+{\mathrm{\beta }}_{\mathrm{VS}}{\mathrm{C}}_{\mathrm{eq}}}$$E34

3.3.14 Unilan isotherm model

The application of the local Langmuir isotherm and uniform energy distribution is assumed for the Unilan isotherm model [44]. Eq. (35) presents the Unilan isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }}{2{\mathrm{\beta }}_{\mathrm{U}}}ln\left[\frac{1+{\mathrm{K}}_{\mathrm{U}}{\mathrm{C}}_{\mathrm{eq}}{\mathrm{e}}^{{\mathrm{\beta }}_{\mathrm{U}}}}{1+{\mathrm{K}}_{\mathrm{U}}{\mathrm{C}}_{\mathrm{eq}}{\mathrm{e}}^{-{\mathrm{\beta }}_{\mathrm{U}}}}\right]$$E35

The higher the model exponent β_U, the system is more heterogeneous. If β_U is equal to 0, the Unilan isotherm model becomes the classical Langmuir model as the range of energy distribution is zero in this limit [50, 55, 56].

3.3.15 Holl-Krich isotherm model

The Langmuir Isotherm [57] is a version of the Holl-Krich Isotherm Model. The Freundlich isotherm is formed when the concentration of the solvent is low [22]. The Holl-Krich Isotherm Model may be seen in Eq. (36).

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{K}}_{\mathrm{HK}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{HK}}}}{1+{\mathrm{K}}_{\mathrm{HK}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{HK}}}}$$E36

3.3.16 Langmuir-Jovanovic isotherm model

This empirical model is the combined form of both Langmuir and Jovanovic isotherm [58]. The Langmuir-Jovanovic model is given in Eq. (37).

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{C}}_{\mathrm{eq}}\left[1-{\mathrm{e}}^{\left({\mathrm{K}}_{\mathrm{LJ}}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{n}}_{\mathrm{LJ}}}\right)}\right]}{1+{\mathrm{C}}_{\mathrm{eq}}}$$E37

3.4 Four parameter models

The four parameter models discussed in this study are Fritz-Schlunder-IV, Baudu, Weber-van Vliet and Marczewski-Jaroniec models.

3.4.1 Fritz-Schlunder-IV isotherm model

Fritz-Schlunder IV model is another model comprised of four-parameter with combine features of Langmuir-Freundlich isotherm [45]. The model is given in Eq. (38).

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{A}}_{\mathrm{FS}5}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\alpha }}_{\mathrm{FS}5}}}{1+{\mathrm{B}}_{\mathrm{FS}5}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\beta }}_{\mathrm{FS}5}}}$$E38

When the values of α_FS5 and β_FS5 are less than or equal to one, this isotherm is true. The Fritz-Schlunder-IV isotherm transforms into the Freundlich equation at high adsorbate concentrations. If both α_FS5 and β_FS5 are equal to one, the isotherm is reduced to the Langmuir isotherm. This isotherm model becomes the Freundlich at large concentrations of adsorbate in the liquid-phase.

3.4.2 Baudu isotherm model

The Baudu isotherm model was created in response to a disagreement in computing the Langmuir constant and coefficient from slope and tangent across a wide range of concentrations [59]. The Langmuir isotherm model has been modified into the Baudu isotherm model. Eq. (39) explains it.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{b}}_{\mathrm{o}}{\mathrm{C}}_{\mathrm{eq}}^{\left(1+\mathrm{x}+\mathrm{y}\right)}}{1+{\mathrm{b}}_{\mathrm{o}}{\mathrm{C}}_{\mathrm{eq}}^{\left(1+\mathrm{x}\right)}}$$E39

This model is only applicable in the range of (1 + x + y) < 1 and (1 + x) < 1. For lower surface coverage, Baudu model reduces to the Freundlich equation [58], i.e.:

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\mathrm{m}0{\mathrm{b}}_{\mathrm{o}}{\mathrm{C}}_{\mathrm{eq}}^{\left(1+\mathrm{x}+\mathrm{y}\right)}}}{1+{\mathrm{b}}_0}$$E40

3.4.3 Weber-van Vliet isotherm model

With four parameters, the Weber and van Vliet isotherm model is used to represent equilibrium adsorption data [60, 61, 62]. Eq. (41) depicts the model.

$${\mathrm{C}}_{\mathrm{eq}}={\mathrm{P}}_1{\mathrm{q}}_{\mathrm{eq}}^{\left({\mathrm{P}}_2{\mathrm{q}}_{\mathrm{eq}}^{{\mathrm{P}}_3}+{\mathrm{P}}_4\right)}$$E41

Multiple nonlinear curve fitting approaches based on the reduction of the sum of squares of residuals can be used to define the isotherm parameters P₁, P₂, P₃, and P₄.

3.4.4 Marczewski-Jaroniec isotherm model

The Marczewski-Jaroniec isotherm model is analogous to the Langmuir isotherm model [62, 63]. Eq. (42) represents the Marczewski-Jaroniec isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}={\mathrm{q}}_{\max }\left[\frac{{\left({\mathrm{K}}_{\mathrm{MJ}}{\mathrm{C}}_{\mathrm{eq}}\right)}^{{\mathrm{n}}_{\mathrm{MJ}}}}{1+{\left({\mathrm{K}}_{\mathrm{MJ}}{\mathrm{C}}_{\mathrm{eq}}\right)}^{{\mathrm{n}}_{\mathrm{MJ}}}}\right]$$E42

The spreading of distribution along the route of increasing adsorption energy is described by K_MJ.

3.5 Five parameter model

Accounting for the high parameter models offers unmistakable information on the process of adsorption under equilibrium conditions. Only one five-parameter model, the Fritz-Schlunder-V isotherm model, is used in this section.

3.5.1 Fritz-Schlunder-V isotherm model

The Fritz-Schlunder adsorption isotherm model was created with the goal of more precisely reproducing model modifications for applicability over a wide range of equilibrium data [45]. Eq. (43) represents the Fritz-Schlunder adsorption isotherm model.

$${\mathrm{q}}_{\mathrm{eq}}=\frac{{\mathrm{q}}_{\max }{\mathrm{K}}_{1\mathrm{FS}5}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\alpha }}_{\mathrm{FS}5}}}{1+{\mathrm{K}}_{2\mathrm{FS}5}{\mathrm{C}}_{\mathrm{eq}}^{{\mathrm{\beta }}_{\mathrm{FS}5}}}$$E43

4. Results and discussion

The findings of the experiments suggest that the highest Ni(II) adsorption by BGMA may be attained at a pH of 6 and a biomass loading of 2 g of BGMA. BGMA has an adsorption capability of 42.056 mg/g. During the continuous 24 hours of contact time, the agitation rate of 120 rpm is maintained.

Figure 1 depicts the visual effect of Ni(II) starting metal ion concentration on equilibrium metal absorption and % clearance. The largest percent elimination of Ni(II) metal ions is observed at low starting metal ion concentrations. The diminishing trend in Ni(II) metal ion removal is observed with an increase in initial metal ion concentration due to an increase in the ratio of the initial number of metal ions to the fixed number of active sites. Furthermore, for a certain number of active sites, the amount of substrate metal ions accommodated in the interlayer gap rises, resulting in decreased metal ion removal. An increase in the initial metal ion concentration causes a decrease in the ionic strength of the solution, which helps to improve metal absorption. As a result of the lowering ionic strength, a rise in initial metal ion concentration raises the equilibrium metal uptake (q_eq).

Figure 1.

Effect of initial metal ion concentration of equilibrium metal uptake and % removal for adsorption of Ni(II) onto BGMA.

4.1 FTIR-characterisation of BGMA biomass

Figure 2 depicts an FTIR spectroscopic investigation of BGMA prior to Ni(II) adsorption. The ▬NH stretching is shown by the wide adsorption bands at 3696.36 cm⁻¹, 3620.77 cm⁻¹, and 3408.94 cm⁻¹. The ▬CH₂ stretching is measured at 2928.18 cm⁻¹. The wide adsorption band at 1643.81 cm⁻¹ might be attributed to the carboxylic C〓O group, whereas the carboxylate group is represented by the adsorption band at 1427.97 cm⁻¹. Furthermore, the band at 1039.87 cm⁻¹ shows C▬N amide stretching. Figure 3 depicts an FTIR spectroscopic investigation of BGMA following Ni(II) adsorption. The shifts of peaks at 3695.90–34696.36 cm⁻¹, 1643.81–1644.54 cm⁻¹, and 1041.07–1039.87 cm⁻¹ after Ni(II) adsorption indicate that the amide ▬NH bonding, CH stretching, carboxylic acid, and hydroxyl groups are the main functional groups involved in the adsorption of Ni(II) metal ions.

Figure 2.

FTIR spectra of BGMA biomass after adsorption of Ni(II).

Figure 3.

FTIR spectra of BGMA biomass before adsorption of Ni(II).

4.2 Adsorption isotherms

Figure 4 depicts the experimental adsorption behaviour of Ni(II) from its synthetic aqueous solutions onto BGMA, which is particularly important in distinguishing the form of the isotherm [32, 64]. According to Giles et al. [65], the isotherm of Ni(II) onto BGMA is detected as an L curve pattern. As a result, it is determined that there is no considerable rivalry between the solvent and the adsorbate for the active sites of BGMA. Also According to Limousin et al., [66], BGMA has a restricted sorption capability for Ni(II) adsorption at the circumstances used in this study.

Figure 4.

Experimental results of adsorption of Ni(II) onto BGMA.

4.2.1 One parameter model

The experimental data for the adsorption of Ni(II) onto BGMA is fitted to Henry’s law (one parameter) model. This model’s parameter values and regression coefficient R² are shown in Table 2. The model fails to match the experimental data under equilibrium conditions due to the small R² value.

Model	Parameter	Value	SSE	R^2	RMSE
Henry’s law model	K	1.132	274.4	0.8001	5.522

Table 2.

Parameter values of one parameter (Henry’s law) model for adsorption of Ni(II) onto BGMA.

4.2.2 Two parameter models

Table 3 shows the regression coefficients and parameter values of two parameter adsorption isotherm models for Ni(II) adsorption onto BGMA. The R² values for the Dubinin-Radushkevich, Hill-de Boer, Fowler-Guggenheim, Halsey, Harkin-Jura, Elovich, and Kiselev models are weak and negative when compared to the experimental data. Though the R² values of the Temkin and Flory-Huggins models suggest their relevance, the parameter values found in both models (b_T and n_FH) appear to be too high and negative, which are not physically realisable.

S. no.	Model	Parameter	Value	SSE	R^2	RMSE
1	Henry’s law with intercept model	K	0.8521	58.67	0.9573	2.708
		m	7.926
2	Langmuir isotherm model	bL	0.05284	83.52	0.9432	3.231
		qmax	55.75
		RL	0.2534
3	Freundlich isotherm model	aF (mg/g)	5.029	45.44	0.9669	2.383
		nF	1.783
4	Dubinin-Radushkevich model	BDR	0.5688	7027	−4.118	29.64
		KDR	0.1927
5	Temkin model	AT	0.7986	97.87	0.9287	3.498
		bT	240
6	Hill-de Boer model	K1	2.024 × 10^4	1.294 × 10^4	−1.401	40.22
		K2	1.535 × 10^5
7	Fowler-Guggenheim model	KFG	4.692 × 10^−8	559.2	0.8963	8.36
		W	−1.067
8	Flory-Huggins isotherm	KFH	0.0008136	5092	0.9012	25.23
		nFH	−1.067
9	Halsey isotherm model	KHa	1.91 × 10^4	3907	−1.845	22.1
		nHa	2.855
10	Harkin-Jura isotherm model	AHJ	1.379	23	−5.9015	23
		BHJ	2.559
11	Jovanovic isotherm model	KJ	0.05745	16.9	0.9359	3.655
		qmax	42.04
12	Elovich isotherm model	KE	−4.5 × 10^−18	1.66 × 10^6	−1210	455.8
		qmax	0.9139
13	Kiselev isotherm model	KeqK	85.68	7027	−4.118	29.64
		KnK	85.64

Table 3.

Parameter values of two parameter adsorption isotherm models for adsorption of Ni(II) onto BGMA.

Only four models, namely Freundlich, Henry’s law with intercept, Jovanovic, and Langmuir, are considered to carry out the following discussion. Figure 5 depicts a plot of C_eq (mg/L) vs. q_eq (mg/g) for the two models, as well as experience data.

Figure 5.

Comparison of experimental values of equilibrium uptake of Ni(II) with two parameter model values.

With experimental data, the Freundlich isotherm model performs better. Its R² score indicates its relevance to a large extent. Adsorption sites are stimulated via the surface exchange process, resulting in enhanced adsorption. Because the value of n_F is in the 1–10 range, It means that Ni(II) adsorption from its synthesised solution onto BGMA is favourable. The value of 1/n_F is calculated as 0.5608, which is closer to zero, assuring that the active sites of BGMA for Ni(II) adsorption on its surface are more heterogeneous.

Henry’s law with intercept model has a high R² value, implying its importance. The incorporation of the intercept term considerably improves the linear connection between q_eq and C_eq.

Langmuir gives improved agreement (R² = 0.9432) with experimental adsorption data when followed by the Freundlich and Henry’s law with intercept model. It denotes monolayer coverage of the Ni(II) at the BGMA’s outer surface. The value of b_L is 0.05284 mL/g, which quantifies the affinity of Ni(II) and BGMA. The computed value of R_L is 0.2534, indicating that the adsorption of Ni(II) onto BGMA is favourable.

However, the q_max of BGMA calculated by this model (55.75 mg/g) differs from the observed q_max value (42.056 mg/g). The variation cannot be significant. The concordance between experimental adsorption data and the Jovanovic isotherm model is quite substantial. It is demonstrated by its R² value (0.9359) and q_max value (42.04 mg/g).

Because the R² values of the four models are high and provide strong mathematical agreement with the experimental results, it cannot be stated that the four isotherms or processes are suitable for adsorption of Ni(II) onto BGMA across the whole concentration range studied. Figure 6 confirms this by comparing the four models to experimental equilibrium metal uptake and demonstrating the amount of concordance.

Figure 6.

Concurrence of two parameter model values of equilibrium uptake of Ni(II) with experimental values.

The Freundlich isotherm mechanism clearly indicates maximum satisfaction with the equilibrium experimental data based on the R², SSE, and RMSE values.

4.2.3 Three parameter models

Table 4 shows the parameter values for three parameter adsorption isotherm models for the adsorption of Ni(II) onto BGMA. Hill, Redlich-Peterson, Langmuir-Freundlich, Fritz-Schlunder-III, Radke-Prausnits, and Jossens isotherm models have low R² values. However, the models Sips, Radke-Prausnits -I, Radke-Prausnits -III, Toth, Khan, Koble-Corrigan, Jovanovic-Freundlich, Brouers-Sotolongo, Vieth-Sladek, Unilan, Holl-Krich, and Langmuir-Jovanovic isotherm models demonstrate their relevance by strong R² values, however the parameter and q_max (mg/g) values produced are either negative or excessively high, implying that they are not physically realisable. As a result, all 16 models are dropped from the ongoing debate.

S. no.	Model	Parameter	Value	SSE	R^2	RMSE
1	Hill isotherm model	KH	423.4	350.2	0.745	7.073
		nH	0.2637
		qmax	5630
2	Redlich-Peterson isotherm model	ARP	1.333 × 10^4	188.3	0.8628	5.187
		BRP	7939
		β	0.1181
3	Sips isotherm model	KS	0.002087	49.55	0.9639	2.661
		qmax	2749
		β	0.5235
4	Langmuir-Freundlich model	KLF	7.23	1613	0.07953	15.18
		mLF	0.657
		qmax	26.28
5	Fritz-Schlunder-III isotherm model	KFS3	0.8637	274.7	0.7999	6.265
		MFS3	−8.958
		qmax	1.31
6	Radke-Prausnits isotherm model-I	KRaP1	115.2	50.34	0.9633	2.682
		MRaP1	0.39
		qmax	0.2358
	Radke-Prausnits isotherm model-II	KRaP2	5300	158.3	0.8847	4.756
		MRaP2	0.1612
		qmax	1.953
	Radke-Prausnits isotherm model-III	KRaP3	0.0248	45.5	0.9669	2.549
		MRaP3	0.5569
		qmax	206.7
7	Toth isotherm model	KT	0.9765	70.13	0.9489	3.165
		nT	0.06976
		qmax	1.614 × 10^5
8	Khan isotherm model	aK	0.4462	45.39	0.9669	2.546
		bK	6.611
		qmax	1.814
9	Koble-Corrigan isotherm model	AKC	5.277	39.14	0.9715	2.364
		BKC	−0.169
		nKC	0.3292
10	Jossens isotherm model	bJ	−0.2596	483	0.6482	8.307
		J	10,710
		KJ	4789
11	Jovanovic-Freundlich isotherm model	KJF	0.00255	46.92	0.9658	2.589
		nJF	0.5391
		qmax	2133
12	Brouers-Sotolongo isotherm model	KBS	0.003327	49.44	0.964	2.658
		αBS	0.523
		qmax	1724
13	Vieth-Sladek isotherm model	KVS	0.6782	33.05	0.9759	2.173
		βVS	0.3985
		qmax	14.37
14	Unilan isotherm model	KU	10.59	49.87	0.9637	2.669
		βU	−3.1
		qmax	−0.08062
15	Holl-Krich isotherm model	KHK	0.002149	49.6	0.9639	2.662
		nHK	0.5234
		qmax	2671
16	Langmuir-Jovanovic isotherm model	KLJ	−0.009672	52.96	0.9614	2.751
		nLJ	0.4409
		qmax	820.8

Table 4.

Parameter values of three parameter isotherm models for adsorption of Ni(II) onto BGMA.

4.2.4 Four parameter models

Table 5 shows the parameter and R² values for four parameter isotherm models. To understand the adsorption mechanism of Ni(II) onto BGMA, the Fritz-Schlunder-IV isotherm model, Baudu isotherm model, Weber-van Vliet isotherm model, and Marczewski-Jaroniec isotherm model are investigated.

Model	Parameter	Value	SSE	R^2	RMSE
Fritz-Schlunder-IV isotherm model	AFS5	5.188 × 10^−5	110.7	0.9194	4.296
	BFS5	−1
	αFS5	1.092
	βFS5	−1.58 × 10^−5
Baudu isotherm model	x	3.491	42.03	0.9694	2.647
	y	0.5412
	bo	0.3431
	qmax	5.373
Weber-van Vliet isotherm model	P1	29.91	1373	−1.861 × 10^−10	1.471
	P2	−6.569
	P3	8.425
	P4	0.795
Marczewski-Jaroniec isotherm model	KMJ	16.93	652	0.5252	10.42
	mMJ	11.11
	nMJ	0.7926
	qmax	27.26

Table 5.

Parameter values of four parameter isotherm models for adsorption of Ni(II) onto BGMA.

The Baudu and Fritz-Schlunder-IV models, which have higher R² values than the Weber-van Vliet and Marczewski-Jaroniec models, are the most significant of the four models. Unfortunately, the exponents and parameters of all four models are either zero, very low, or excessively high, making them physically impossible to realise. Like a result, just as in the case of the three parameter models, all four parameter models fail to describe the process of adsorption, and the discussion is unnecessary.

4.2.5 Five parameter model

Table 6 shows the Fritz-Schlunder-V model’s parameter values. The R² value denotes its relevance. Figure 7 depicts a comparison of experimental Ni(II) metal uptake with the Fritz-Schlunder-V model under equilibrium conditions. Figure 8 depicts the agreement of Fritz-Schlunder-V parameter model results for equilibrium Ni(II) uptake with experimental data.

S. no.	Model	Parameter	Value	SSE	R^2	RMSE
1	Fritz-Schlunder-5 isotherm model	K1FS5	0.3518	45.45	0.9669	3.015
		K2FS5	1.927
		αFS5	0.5605
		βFS5	0.0001938
		qmax	41.89

Table 6.

Parameter values of five parameter isotherm model for adsorption of Ni(II) onto BGMA.

Figure 7.

Comparison of experimental values of equilibrium uptake of Ni(II) with five parameter model values.

Figure 8.

Concurrence of five parameter model values of equilibrium uptake of Ni(II) with experimental values.

The q_max value of 41.89 mg/g for this isotherm model is extremely similar to the experimental q_max value of 42.056 mg/g. As a consequence, the Fritz-Schlunder-V isotherm model is firmly established for the adsorption of Ni(II) metal ions from synthetic aqueous solution onto BGMA.

5. Conclusion

The adsorption of Ni(II) metal ions from synthetic aqueous solutions is investigated using BGMA as a low-cost adsorbent. At a pH of 6, 2 g of biomass input, and an agitation speed of 120 rpm, the greatest adsorption capacity of BGMA was determined to be 42.056 mg/g. Because the ionic strength decreases with increasing initial Ni(II) metal ion concentration, the percentage elimination decreases and the equilibrium metal absorption (q_eq) increases. The equilibrium experimental data suggests that the isotherm has a L shape, indicating that solvent and Ni(II) are competing for the active sites of BGMA. Furthermore, it suggests that the BGMA has a restricted capacity for Ni adsorption(II). Furthermore, the efficacy of various isotherms for modelling is investigated using a 1-parameter isotherm, a 13-parameter isotherm, a 16-3-parameter isotherm, a 4-4-parameter isotherm, and a 1-5-parameter isotherm. The experiences are graphically depicted. The Fritz-Schlunder-V isotherm model is obviously relevant in characterising the mechanism of Ni(II) adsorption under the conditions utilised in this work, which was followed by Freundlich. The q_max of 41.89 mg/g for this model reveals its significance even more clearly.

Nomenclature

A	Fritz-Schlunder parameter
aF	Freundlich adsorption capacity (L/mg)
AHJ	Harkin-Jura isotherm constant
aK	Kahn isotherm model exponent
AKC	Koble-Carrigan’s isotherm constant
ARP	Redlich-Peterson isotherm constant (L/g)
AT	Temkin equilibrium binding constant corresponding to the maximum binding energy
B	Fritz-Schlunder parameter
BGMA	blue green marine algae
b	Langmuir constant related to adsorption capacity (L/mg)
b0	Langmuir isotherm equilibrium constant
BDR	Dubinin-Radushkevich model constant
BHJ	Harkin-Jura isotherm constant
bJ	Jossens isotherm model parameter
bK	Khan isotherm model constant
BKC	Koble-Carrigan’s isotherm constant
bL	Langmuir constant related to adsorption capacity (mg/g)
BRP	Redlich-Peterson isotherm constant (L/mg)
bT	Temkin constant which is related to the heat of sorption (J/mol)
C	Henry’s law model intercept
Ceq	concentration of adsorbate in bulk solution at equilibrium (mg/L)
Cin	initial adsorbate concentration (mg/L)
J	Jossens isotherm model parameter
K	Henry’s constant
K1	Hill-de Boer constant (L/mg)
K1FS5	Fritz-Schlunder-V parameter
K2	energetic constant of the interaction between adsorbed molecules (kJ/mol)
K2FS5	Fritz-Schlunder-V parameter
KBS	Brouers-Sotolongo model isotherm parameter
KDR	Dubinin-Radushkevich model uptake capacity
KE	Elovich constant (L/mg)
KFG	Fowler-Guggenheim equilibrium constant (L/mg)
KFH	Flory-Huggins equilibrium constant (L/mol)
KFS3	Fritz-Schlunder III equilibrium constant (L/mg)
KH	Hill isotherm constant
KHa	Halsey isotherm constant
KHe	Henry’s constant
KHK	Holl-Krich isotherm model parameter
KJ	Jossens isotherm model parameter
KJ	Jovanovic constant
KJF	Jovanovic-Freundlich isotherm equilibrium constant
KK	Kiselev equilibrium constant (L/mg)
KLF	Langmuir-Freundlich equilibrium constant for heterogeneous solid
KLJ	Langmuir-Jovanovic model parameter
KMJ	Marczewski-Jaroniec isotherm model parameter that characterise the heterogeneity of the adsorbent surface.
KnK	equilibrium constant of the formation of complex between adsorbed molecules
KRaP	Radke-Prausnits equilibrium constant
KS	Sips isotherm model constant (L/mg)
KT	Toth isotherm constant (mg/g)
KU	Unilan isotherm model parameter
KVS	Vieth-Sladek isotherm model parameter related to Henry’s law
mFS3	Fritz-Schlunder III model exponent
mLF	Langmuir-Freundlich heterogeneity parameter
mRaP	Radke-Prausnits model exponent
nF	Freundlich adsorption intensity
nFH	number of adsorbates occupying adsorption sites
nH	exponent of Hill adsorption model
nHa	Halsey isotherm exponent
nHK	Holl-Krich isotherm model exponent
nJF	Jovanovic-Freundlich isotherm exponent
nKC	Koble-Carrigan’s isotherm constant
nLJ	Langmuir-Jovanovic model exponent
nMJ	Marczewski-Jaroniec isotherm model parameter that characterise the heterogeneity of the adsorbent surface
nT	Toth isotherm exponent
P1	Weber and van Vliet isotherm model parameter
P2	Weber and van Vliet isotherm model parameter
P3	Weber and van Vliet isotherm model parameter
P4	Weber and van Vliet isotherm model parameter
qeq	amount of adsorbate in adsorbent at equilibrium (mg/g)
qmax	maximum quantity of solute adsorbed by the adsorbent (mg/g)
R	gas constant (8.314 J/mol K)
RL	Langmuir separation factor
T	absolute temperature (K)
W	interaction energy between adsorbed molecules (kJ/mol)
x	Baudu isotherm model parameter
y	Baudu isotherm model parameter
θ	fractional surface coverage
βRP	Redlich-Peterson isotherm exponent
βS	Sips isotherm exponent
αBS	Brouers-Sotolongo model isotherm parameter is related to adsorption energy
βVS	Vieth-Sladek isotherm model parameter related to Langmuir
βU	Unilan isotherm model exponent
αFS5	Fritz-Schlunder-V parameter
β2FS5	Fritz-Schlunder-V parameter