*3.3. Kinetic Models*

The pseudo-first-order, pseudo-second-order, and intraparticle diffusion models were engaged to evaluate the adsorption kinetics of the Fe–Cu/Alg–LS nanocomposite. [37]. The optimum situations were conventional as pH 6 for (CIP) 7 for (LEV), Fe–Cu/Alg–LS G nanocomposite mass of 0.2 g/25 mL, a contact time of (40 min (CIP), 45 min (LEV)), and 20 ppm (CIP), 10 ppm (LEV) as the initial concentrations.

#### 3.3.1. Pseudo-First-Order Reaction Kinetics

Figure 7A [42] provides a description of the PFOR reaction kinetics equation. The following equation is used to represent the current starting phase:

$$\text{Log}\left(\mathbf{q}\_{\text{e}}-\mathbf{q}\_{\text{l}}\right)-\text{log}\,\mathbf{q}\_{\text{e}}=-\,\text{K}\_{\text{ads}}\,\text{t}/2.503\tag{3}$$

where qt (mg/g) represents the adsorption capacity at time t. Kads (min−1) stands for the rate constant of PFOR adsorption.

**Figure 7.** Adsorption kinetics: (**A**) pseudo-first-order reaction (PFORE), (**B**) pseudo-second-order reaction (PSORE), (**C**) Morris–Weber equation for CIP and LEV on the Fe–Cu/Alg–LS nanocomposite (sorption time, (40 min (CIP) 45 min. (LEV) by 0.2 g/25 mL of the nanocomposite at pH 6 (CIP) and 7 (LEV).

In this study, a linear relationship was recognized for the adsorption of CIP and LEV ions onto the Fe–Cu/Alg–LS nanocomposite. The values of qe and kads were measured from the slope and intercept by plotting log (qe−qt) versus t. The PFOR kinetics are illustrated in Figure 7A. The outcomes exhibited correlation coefficients (R<sup>2</sup> = 0.8073, 0.9613) for CIP and LEV. The collected data show that the pseudo-first-order model has a poor fit for the adsorption of CIP and LEV onto the Fe–Cu/Alg–LS nanocomposite.

#### 3.3.2. Pseudo-Second-Order Reaction

The PSOR kinetic model [43] is illustrated in the following equation:

$$\text{t/q} = 1/\text{K}\_2\text{q}\_{\text{e2}} + \text{t/q}\_{\text{fe}} \tag{4}$$

The PSOR rate constant, denoted by K2 (g/mg/min), is shown in Figure 7B. When t/qt is plotted versus t, the slopes and intercepts determine the values of the rate constant K2, equilibrium adsorption capacity qe, and correlation coefficient (R2).

PSOR correlation coefficients (R2) for the Fe–Cu/Alg–LS nanocomposite in Table 3 maintained high values. The outcomes exhibited high correlation coefficients (R<sup>2</sup> = 1) for both CIP and LEV. The statistics imply that the CIP and LEV adsorption suitable for the pseudo-second-order kinetics.


**Table 3.** Kinetic modeling with the PFOR, PSOR and Morris–Weber equations.

#### 3.3.3. Morris–Weber Kinetic Equation

The Morris–Weber Equation (5) [44] can be used to represent the intraparticle mass transfer diffusion, as shown in Figure 7C.

$$\mathbf{q} = \mathbf{K}\_{\mathbf{d}} \text{ (t)}^{1/2} \tag{5}$$

where the uptake of metal ions is denoted by the symbol q (g/g), the intraparticle mass transfer diffusion rate constant is denoted by Kd, and the square root of time is denoted by the symbol t1/2. Only in the shorter stage, if the intraparticle diffusion and adsorption data overlapped, would it occur. The first part is linear, which is related to the boundary layer effect, as shown by the Morris–Weber equation in Figure 7C. However, the intraparticle diffusion effect may be related to the second component [45]. The fact that practically all sorption occurs within the first 40 min for CIP and 45 min for LEV, with a clear linear trend, indicates that the porosity of nanocomposites exceeds the resistance influencing intraparticle diffusion [46]. For CIP and LEV adsorption, the intraparticle diffusion rate constant value Kd was estimated to be 0.0831 and 0.0727 (g/g min−1), respectively, onto the Fe–Cu/Alg–LG nanocomposite, suggesting CIP and LEV ions move to the composite. The values of Kd for both antibiotics represent the rate of diffusion of pollutants towards the pore of nanocomposite, accordingly the rate of diffusion of CIP is higher than LEV onto nanocomposite. The kinetic modeling with the PFOR, PSOR and Morris–Weber equations are detailed in Table 3.

#### *3.4. Isotherm Model*

To adequately understand the adsorption process, isotherm studies are required [47]. The Langmuir, Freundlich, and Dubinin–Radushkevich isotherm models were used to study the adsorption process. The Fe–Cu/Alg–LS nanocomposite was 0.2 g/25 mL in mass, with contact times of 40 min for CIP (20 ppm) and 45 min for LEV (10 ppm) according to the optimized experimental conditions.

The Langmuir isotherm was used to explain the adsorption of any substance on a homogeneous surface with minimal interaction between the molecules that had been adsorbed [48]. The model assumes a homogeneous uptake in accordance with the saturation level of the monolayer on the surface with the highest adsorption. The following gives an illustration of the Langmuir linear equation model [49]:

$$\text{Ce}/\text{q}\_{\text{e}} = 1/\text{K}\_{\text{L}} \text{ q}\_{\text{max}} + (1/\text{q}\_{\text{max}}) \cdot \text{C}\_{\text{e}} \tag{6}$$

where KL (L·mg−1) denotes the monolayer's maximum adsorption capacity and qmax (mg.g−1) denotes its maximum capacity for sorption heat. Figure 8A,B illustrate the Langmuir adsorption isotherm that was constructed on the basis of monolayer adsorption through the adsorption process. The equilibrium absorption of the homogeneous surface of the adsorbents is explained by the Langmuir model.

**Figure 8.** (**A**) Langmuir (**B**) Freundlich adsorption isotherm.

The Freundlich model, particularly for heterogeneous surfaces [50,51], is one of the first empirical equations compatible with the exponential distribution of active centers as follows:

$$
\ln \mathbf{q}\_{\mathbf{e}} = \ln \mathbf{K}\_{\mathbf{f}} + 1/n \ln \mathbf{C}\_{\mathbf{e}} \tag{7}
$$

If Kf denotes adsorption capacity, n denotes intensity, and Kf is a crucial and relative indicator of adsorption capacity; it denotes a beneficial adsorption extent. Adsorption is considered suitable when n is greater than 1 [52]. The results demonstrate that the Langmuir model performed better than the Freundlich model in describing the experimental data of the Fe–Cu/Alg–LS nanocomposites. The values for the correlation coefficient (R2) are described in Table 3. For both CIP and LEV adsorption, the R2 values from the Langmuir model data were 0.9731 and 0.9990, above those of the Freundlich isotherm. The displacement of CIP and LEV ions appears to be a monolayer covered on the surface of the Fe–Cu/Alg–LS nanocomposite, according to the adsorption results. As a result, the outcomes closely matched the Langmuir model.

Dubinin–Radushkevich Isotherm

This model fits exceedingly well with the Gaussian energy distribution and adsorption techniques that were used on a heterogeneous surface. The D-R equation is as follows [53]:

$$\ln \mathbf{q} = \ln \mathbf{q}\_{\text{(D-R)}} - \mathcal{B}\varepsilon^2 \tag{8}$$

$$
\varepsilon = \text{RT} \ln(1 + 1/\mathcal{C}\_{\text{e}}) \tag{9}
$$

When the ideal gas constant, R, is taken into account, the theoretical adsorption capacity, q(D-R) (mg·g<sup>−</sup>1), the activity coefficient, ß (mol2 kJ<sup>−</sup>2), the Polanyi potential(ε), T (absolute temperature in K), and E (kJ mol−1), represented as the free energy change, are as follows:

$$\mathbf{E} = 1/(\mathbf{2}\mathbf{8})^{1/2} \tag{10}$$

The E value can be used to identify the type of reaction. Physical forces are predicted to have an impact on the adsorption process if E < 8 kJ mol−1. If E is between 8 and 16 kJ mol−1, chemical ion exchange takes place during the sorption process. Particle diffusion may also be used to determine the sorption process if E is more than 16 kJ mol−<sup>1</sup> [54]. Table 4 provides a list of the D-R model simulation data. E values for the absorption of CIP and LEV ions onto the Fe–Cu/Alg–LS nanocomposite were 0.7624 and 0.7446 kJ mol<sup>−</sup>1. As a result, if E is less than 8 kJ mol<sup>−</sup>1, physical adsorption will affect the sorption [55].

**Table 4.** Sorption isotherms.

