1. Introduction

2. Materials and Methods

3. Results and discussion

3.3. Adsorption kinetic models

The adsorption process consists in three steps:

-

External diffusion which means the mass transfer of the adsorbate from the bulk solution to the external surface of the adsorbent;

-

Internal diffusion of the adsorbate through the pores of the adsorbent;

-

The adsorption itself on the active centers of the adsorbent.

The slowest step among the three steps is the rate limiting step. This can be elucidated by fitting different kinetic models to establish the most probable adsorption mechanism. The experimental data on adsorption kinetics were examined using a variety of kinetic models, pseudo-first-order model, pseudo-second-order model, Elovich model and intraparticle diffusion model [20].

3.3.1. Pseudo-first order kinetics model

The kinetics equation of pseudo-first order model and its linearized form may be represented as follow [20,33,34,35,36,37]:

$$\frac{d{q}_t}{dt}=k_1\left(q_e-q_t\right) \left(non-linear form\right)$$

(3)

$$\mathit{ln}\left(q_e-q_t\right)=\mathit{ln}{q}_e-k_{1}t \left(linear form\right)$$

(4)

Where k₁ (min⁻¹) is the rate constant for the pseudo-first order kinetics model, q_e (mg/g), q_t (mg/g) are the amounts of Cu²⁺ and Ni² cations retained on weight unit of adsorbent at equilibrium, and at any time t (min), respectively.

The plot of ln (q_e-q_t) versus contact time t for Alginate-Clay bio-nanocomposite beads, gives a straight line of slope –k₁ and intercepts lnq_e (Figure 9). The values of the theoretical adsorption capacity (q_{e, the}), the rate constant for the pseudo-first order kinetics model (k₁) and the correlation coefficient (R²) are presented in Table 2. The table shows that the value of theoretical adsorbed amount q_e is not quite similar to the experimental value (q_{e, the} = 20.46 mg/g < q_{e, exp} = 72.82 mg/g for Ni²⁺ and q_{e, the} = 2.55 mg/g < q_{e, exp} = 83.30 mg/g for Cu²⁺, suggesting the insufficiency of pseudo-first-order model. We find that, under these conditions, the pseudo-first order model is not adequate to describe the adsorption kinetics of Cu²⁺ and Ni² cations from aqueous solutions onto Alginate-Moroccan clay bio-nanocomposite beads.

3.3.2. Pseudo- second order kinetics model

The rate equation and its linearized form for the pseudo-second-order kinetics model can be represented as follows [20,33,34,35,36,37,38]:

$$\frac{d{q}_t}{dt}=k_2{\left(q_e-q_t\right)}^2 \left(non-linear form\right)$$

(5)

$$\frac{t}{q_t}=\frac{1}{k_2{q}_e^2}+\left(\frac{1}{q_e}\right)t \left( linear form\right)$$

(6)

Where, k₂ (g.mg-1min^-1) is the rate constant for the pseudo-second order kinetics model, q_e (mg/g), q_t (mg/g) are the amounts of Cu²⁺ and Ni² cations retained on weight unit of adsorbent at equilibrium, and at any contact time t (min), respectively. The pseudo-second order plots for Cu²⁺ and Ni² cations adsorption are presented in Figure 10, and the kinetic parameters are given in Table 2. The correlation coefficient for the pseudo second order kinetic model is nearly equal to 1, and the value of theoretical adsorption capacity (q_{e, the} ) is comparable to the experimental one (q_{e, the} = 75.75 mg/g for Ni²⁺ and q_{e, the} = 90. 09 mg/g for Cu²⁺).

Therefore, it was concluded that the pseudo-second order adsorption model is more appropriate to describe the adsorption kinetics of Cu²⁺ and Ni² cations on Alginate- Moroccan clay bio-nanocomposite.

3.3.3. Elovich kinetic model

The Elovich model is a useful tool for studying systems with heterogeneous surfaces, especially when describing the kinetics of chemisorption [20]. This model is mathematically expressed through equations 7 and 8. In these equations, 'q_e' and 'q_t' (measured in mg/g) denote the quantities of adsorbed Cu²⁺ and Ni²⁺ cations at equilibrium, and at any specific contact time 't' (in minutes), respectively.

The kinetics equation Elovich model and its linearized form may be expressed as:

$$\frac{d{q}_t}{dt}=\alpha e^{-\beta q_t} \left(non-linear form\right)$$

(7)

$$q_t=\frac{ln\left(\alpha \beta \right)}{\beta }+\frac{1}{\beta }\mathit{ln}(t) \left(linear form\right)$$

(8)

Where, q_e (mg/g) and q_t (mg/g) are the amounts of Cu²⁺ and Ni²⁺ cations adsorbed at equilibrium and at any contact time t (min), respectively. α (mg/g^/min) is the initial adsorption rate and β (g/mg) is the desorption constant related to the extent of the surface coverage and activation energy for chemisorption. The Elovich kinetic constants α and β are obtained from the intercept and the slope respectively (Figure 11). The correlation coefficient indicates that the Elovich model is not adequate to characterize the Cu²⁺ and Ni²⁺ cations adsorption on bio-nanocomposite beads. However, based on the correlation coefficient analysis, it is evident that the Elovich model is not sufficient to accurately characterize the adsorption of Cu²⁺ and Ni²⁺ cations on bio-nanocomposite beads.

3.3.4. Intra-particle diffusion kinetics model

Fitting the experimental data to an intraparticle diffusion model is the most widely used method for determining the mechanism involved in the sorption process. One or more processes, such as boundary layer (film) or external diffusion, diffusion at the surface, internal pore diffusion, or a combination of several steps, may be used to approximate the overall adsorption of solute onto the solid surface [20,39].

To calculate the initial rate of intra-particle diffusion, Equation 9 is linearized. In this equation, k_p (mg^-1·min^1/2) represents the intra-particle diffusion rate constant, c (mg/g) is the concentration of Cu²⁺ and Ni²⁺ cations from the solution at equilibrium, and q_t (mg/g) denotes the amount of Cu²⁺ and Ni²⁺ cations retained on a unit weight of the adsorbent at contact time t (minutes). The graphical representation of this relationship is presented in Figure 12.

$$q_t=k_p{t}^{1/2}+c \left(linear form\right)$$

(9)

The intra-particle diffusion, k_p, values were obtained from the slope of the straight-line part of plot of q_t versus t^1/2, for various solutions temperature. The correlation coefficients (R²) for both studied cations are 0.815 for Ni²⁺ and 0.530 for Cu²⁺ at 25°C. This correlation coefficient indicates that the intraparticle diffusion model is not suitable to describe the kinetics of Cu²⁺ and Ni²⁺ cations adsorption from aqueous solutions on bio-nanocomposite beads. This means that the internal diffusion is either fast and is not the limiting step in the adsorption mechanism or internal diffusion solely is not the limiting step. The values of k_p and c calculated from the slopes and intercepts are summarized in Table 2.

Table 2. Parameters of four kinetic models for Cu²⁺ and Ni²⁺ cations adsorption at C₀=100 mg/L

Table 2. Parameters of four kinetic models for Cu²⁺ and Ni²⁺ cations adsorption at C₀=100 mg/L

Kinetic model	Parameters	Metal Ions
		Ni2+	Cu2+
Pseudo- first order model	qe,exp(mg/g)	72.82	83.30
	K1 (min-1)	0.0059	0.0027
	qe,the(mg/g)	20.46	2.55
	R2	0.737	0.469
Pseudo-Second order model	qe,exp(mg/g)	72.82	83.30
	K2 (g.mg-1min-1)	4.5*10-4	2.5*10-4
	qe,the(mg/g)	75.75	90.09
	R2	0.994	0.992
Elovich model	qe,exp(mg/g)	72.82	83.30
	α	14.64	38.94
	β	0.087	0.090
	R2	0.679	0.806
Intra-particle diffusion	KInt(mg.g-1min-1/2)	1.48	1.35
	CI	50.028	43.52
	R2	0.815	0.530

The bio-nanocomposite utilized in the adsorption investigation of Cu²⁺ and Ni²⁺ cations showed that, for an initial concentration of 100 mg/L, the correlation coefficient (R²) values for the pseudo-second-order adsorption kinetic model were very high (about 0.99). This means that the pseudo-second-order model's estimation of adsorption capacity nearly matched the experimental results. The pseudo-second-order adsorption model is thought to be the best option for explaining the kinetics of Cu²⁺ and Ni²⁺ cations adsorption by the used bio-nanocomposites.

3.4. Isotherm study

Adsorption isotherms, which are important data for understanding the adsorption mechanism, are mathematical models that describe the distribution of adsorbed species between the solid and liquid phases [40]. Several mathematical models have been studied in this work, namely Langmuir, Freundlich, Temkin and Dubinin-Radushkevich (D-R) equations. This study was performed by ranging the initial ion concentration from 100 to 600 mg/L at room temperature.

3.4.1. Langmuir adsorption isotherm

According to the Langmuir adsorption isotherm, homogeneous surfaces are indicated by the solid surface, having a limited amount of identical sites [20,40,41,42,43] The linearized form of the Langmuir equation can be stated as follows:

$$q_e=q_L\frac{K_L{C}_e}{1+K_L{C}_e}\mathrm{}\mathrm{}\mathrm{}(non-linear\mathrm{}form)\mathrm{}\mathrm{}\mathrm{}$$

(10)

$$\frac{1}{q_e}=\frac{1}{q_L}+\frac{1}{q_L{K}_L{C}_e} (linear form)$$

(11)

Where q_e (mg/g) is the amount adsorbed at equilibrium concentration C_e (mg/l), q_L(mg/g) is the Langmuir constant representing maximum monolayer capacity, and K_L (L/mg) is the Langmuir constant related to energy of adsorption.

The plots between 1/q_e and 1/C_e for the adsorption of Cu²⁺ and Ni²⁺ cations are represented in Figure 13. The values of the adsorption capacity (q_L), the Langmuir constant (K_L), and the correlation coefficient (R²) were presented in Table 4. The highest value of the adsorption capacity q_L obtained at 25°C was 370.37 mg/g for Ni²⁺ and 454.54 mg/g for Cu²⁺ ions (Table 4).

The essential feature of the Langmuir isotherm can be expressed by means of ‘R_L’, a dimensionless constant referred to as separation factor or equilibrium parameter, to predict whether an adsorption system is favorable or unfavorable. R_L is calculated using Eq. 12. Where K_L (L/mol) is Langmuir constant and C₀ (mol/L) the highest initial ions concentration.

$$R_L=\frac{1}{1+K_L{C}_0}$$

(12)

The calculated values of parameter R_L for this study were found to be between 0 and 1(0.160 for Ni²⁺ and 0.168 for Cu²⁺ ions), indicating that the adsorption of Cu²⁺ and Ni²⁺ cations onto bio-nanocomposite beads particles was favorable (Table 3).

3.4.2. Freundlich adsorption isotherm

For natural adsorbents which are heterogeneous, the Freundlich equation offers the most suitable adsorption data. The Freundlich adsorption isotherm equation and its linear form can be written as follows [20,40,41,42]:

$$q_e=K_F{C_e}^{1/n} \left(non-linear form\right)$$

(13)

$$\mathit{ln}{q}_e=\mathit{ln}{K}_F+\frac{1}{n}\mathit{ln}{C}_e \left(linear form\right)$$

(14)

where, q_e (mg/g) is the amount of Cu²⁺ and Ni²⁺ cations adsorbed per unit weight of adsorbent; C_e (mg/L) is the equilibrium concentration of solute in the bulk solution; K_F (mg/g) is the Freundlich constant, which is a comparative measure of the adsorption capacity of the adsorbent, and n is an empirical constant related to heterogeneity of the adsorbent surface. The parameter n also indicates the nature of the adsorption process. An adsorption is considered favourable if the value of n is between 0 and 1, unfavourable if its value is more than 1, linear adsorption if its value is 1, and the adsorption process is irreversible if its value is 0. The slope and intercept of the plot ln q_e vs ln C_e were used to obtain the isotherm constants n and K_F (Figure 14). The values for Freundlich constants and correlation coefficients (R²) for both temperatures are also presented in Table 4.

The Freundlich isotherm constants K_F and n are constants incorporating all factors affecting the adsorption process, such as adsorption capacity and intensity of adsorption. The constants K_F and n were calculated from Eq. 14. These experiments confirm the efficiency of the bio-nanocomposite used to remove Cu²⁺ and Ni²⁺ cations from aqueous solutions.

3.4.3. Temkin Isotherm

The Temkin adsorption isotherm model is based on the heat of adsorption of ions, which is due to interactions between adsorbate and adsorbent. The Temkin isotherm equation is written as follows [20,44]:

$$q_e=\frac{RT}{b_T}\mathit{ln}{K}_T{C}_e \left( non-linear form\right)$$

(15)

$$q_e=\frac{RT}{b_T}\mathit{ln}{K}_T+\frac{RT}{b_T}\mathit{ln}{C}_e \left( linear form\right)$$

(16)

Where T is absolute temperature in Kelvin and R the universal gas constant (8.314 J/mol/K). b_T (J/mol) is Temkin isotherm constant related to the heat of adsorption. K_T (L/mg) is the equilibrium binding constant corresponding to the maximum binding energy. The Temkin isotherm plot is presented in Figure 15. The isotherm parameters are given in Table 4. The Temkin constants b_T related to heat of adsorption of Cu²⁺ and Ni²⁺ cations at 25°C were found to be 24.938 J/mol for Ni²⁺ and 21.33 J/mol for Cu²⁺, respectively.

The linear regression of the data points showed rather low R² values (0.829 for Ni²⁺ and 0.865 for Cu²⁺, indicating that adsorption of Cu²⁺ and Ni²⁺ cations did not fully follow the Temkin isotherm.

3.4.4. Dubinin–Radushkevich (D–R) isotherm

When expressing the adsorption mechanism with a Gaussian energy distribution onto a heterogeneous surface, the Dubinin-Radushkevich isotherm is usually used. It is used for calculating the mean free energy of adsorption (E), not the constant adsorption potential or homogenous surface assumption. The D-R equation can be written in both linear and non-linear versions as follows [20]:

$$q_e=q_m{e}^{{-K}_p{\epsilon }^2} \left(non-linear form\right)$$

(17)

$$\mathit{ln}{q}_e=\mathit{ln}{q}_m{-K}_D{\epsilon }^2 \left(linear form\right)$$

(18)

Where q_m (mg/g) is the theoretical saturation capacity and ε is the Polanyi potential that can be calculated from Eq. 19:

$$\epsilon =RT\mathit{ln}\left(1+\frac{1}{C_e}\right)$$

(19)

When the adsorbate is moved from the bulk solution to the surface of the solid, the constant K_D (mol²/J²) provides an estimate of the mean free energy E (kJ/mol) of adsorption per molecule. This may be computed from the K_D value using the following relation (Eq. 20):

$$E=\frac{1}{{\left(2K_D\right)}^{1/2}}$$

(20)

If the value of E is between 8 and 16 kJ/mol, the adsorption process is expected to be chemisorption, while for values of E < 8 kJ/mol, the adsorption process is physical in nature. The results are illustrated in Table 4. The slope of the plot of ln q_e versus ε² gives K_D and the intercept yields the adsorption capacity q_m. As it can be seen in Figure 16 and Table 4, the correlation coefficient values are 0.756 for Ni²⁺ and 0.771 for Cu²⁺, respectively at 25°C. The numerical value of adsorption of the mean free energy is 50 J/mol for Ni²⁺ and 70.71 J/mol for Cu²⁺ (Table 4) corresponds to a physisorption and the predominance of van der Waals forces.

Table 4. Parameters of isotherm models for Cu²⁺ and Ni²⁺ cations. adsorption at 25°C.

Table 4. Parameters of isotherm models for Cu²⁺ and Ni²⁺ cations. adsorption at 25°C.

Model	Parameters	Metal Ions
		Ni2+	Cu2+
Langmuir	qL(mg/g)	370.37	454.54
	KL (L/mg)	0.0087	0.0082
	R2	0.970	0.970
Freundlich	1/n	0.608	0.650
	KF (mg/g)	9.706	9.728
	R2	0.946	0.950
Temkin	KT (L/mg)	0.062	0.066
	bT (J/mol)	24.94	21.33
	R²	0.829	0.865
D-R	KD (mol2/J)	2×10-4	10-4
	qm (mg/g)	229.63	258.60
	E (J/mol)	50	70.71
	R²	0.756	0.771

4. Conclusions

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