#### 3.2.1. Effect of pH

One of the important parameters in the processing of adsorption is the pH of the aqueous media, which directly affects the chemical structure of metal ions and active sites of adsorbents [

27].

Figure 10 shows the effect of various pH values on the adsorption of Pb(II) ions through NC and chitosan. By increasing the pH to 6, the efficiency of removal increased dramatically. Above pH = 6.0, Pb(II) ions precipitated in the hydroxide form [

28,

29]. At pH of 6, the efficiency levels of Pb(II) ion removal for NC and chitosan were 93.88% and 66.32%, respectively. At pH less than 4, the adsorption rate was very low since the adsorbent’s surface was completely covered with H

^{+}, which competed strongly with Pb(II) ions for sites of adsorption, hence the possibility of Pb(II) adsorption was minimized [

30]. In the 4 < pH < 6 range, the H

^{+} ion concentration decreased and Pb(II) ions were more likely to be adsorbed on active sites of NCs or Cs. Hence, for the adsorption of Pb(II) ions, the optimum pH was defined at 6; thus, this pH was selected for all other experiments in this work.

#### 3.2.3. Kinetics of Adsorption

The kinetics of adsorption provides significant data about the reaction paths and the rate of the process. To determine the controlling mechanism for the processing of adsorption, both pseudo-first-order (Equation (3)) and pseudo-second-order (Equation (4)) kinetic models were applied [

32,

33]:

where

q_{t} (mg/g),

q_{e} (mg/g),

k_{1} (min

^{−1}),

k_{2} (g·mg

^{−1}·min

^{−1}), and

t (min) show the amount of metal ion adsorbed, adsorption capacity at adsorption equilibrium, kinetic rate constants for the pseudo-first-order and the pseudo-second-order models, and time of contact, respectively.

In the present investigation, the adsorption data were used to examine the amount of lead ions adsorbed before a contact time, for initial concentrations of metal ions in the range of 10 to 100 mg/L. The uptake of lead ions increased with the contact time and the system reached equilibrium after 60 min for NC and 90 min for chitosan. Increasing metal ion concentration in the aqueous solution increased the chance of an effective collision between the metal ions and the adsorbent and resulted in improved metal ion removal.

Table 1 presents the calculated data for kinetic adsorption of Pb(II) ions that were fitted to Equations (3) and (4). A comparison between the observed and the calculated values of

q_{t} against time revealed a very good fit with the pseudo-second-order rate equation compared with the pseudo-first-order rate equation for both adsorbents. Therefore, the adsorption of lead ions is well illustrated by the pseudo-second-order kinetic model (

Figure 12 and

Figure 13). The constants of pseudo-second-order rate are slightly different for the five initial lead ion concentrations, which reflects the inferior effect of initial concentration. Adoption of the pseudo-second-order kinetic model means that the rate-limiting step may be chemisorption involving valence forces through the sharing or exchanging of electrons between adsorbent and adsorbate [

34]. It was found that the adsorption data fitted well with pseudo-first-order and pseudo-second-order kinetics at high and low initial pollutant concentrations, respectively [

35,

36].

#### 3.2.4. Adsorption Isotherms

The adsorption isotherm model is essential in order to predict and compare the performance of adsorbents. Equilibrium studies have described the affinity and surface properties of adsorbents by constant values and characterized the adsorption capacity of adsorbents [

37,

38,

39,

40,

41]. Langmuir and Freundlich models as adsorption isotherms were studied at concentrations of different initial lead ions in the range of 10–100 mg/L on NC and chitosan (

Figure 14 and

Figure 15). The expressions of the Langmuir isotherm and the linear form of this isotherm are presented in Equations (5) and (6):

where

q_{e} is the adsorption capacity (mg/g) and

C_{e} is the equilibrium absorbent concentration (mg/L), while the maximum adsorption capacity of adsorbents (mg/g) is shown by

q_{m}. The affinity of binding sites is represented by K

_{L} as the Langmuir constant (L/mg) and is a scale of adsorption energy. The Langmuir isotherm model assumes that the maximum adsorption capacity happens on a monolayer of the adsorbent surface. This model states that all adsorption sites have equal energy and that intermolecular forces decrease with increasing distance from the adsorption surface.

The Freundlich isotherm model presumes that adsorption happens at multilayers and various adsorbent sites have different energies. Equations (7) and (8) show the Freundlich isotherm model and its linear form:

where the

n and

K_{F} (mg/g) are Freundlich constants related to the adsorption intensity and adsorption capacity, respectively.

Figure 14a,b presents Langmuir and Freundlich linear fittings, respectively, whereas the isotherm constants are presented in

Table 2. Based on the regression factor (

R^{2}), it was observed that the Langmuir model’s experimental data were better than those of the Freundlich model for both NC and chitosan, implying the predominant occurrence of the monolayer adsorption. The fact that the Langmuir isotherm fitted well to the experimental data may also be due to the homogenous distribution of active sites on NC and chitosan (the Langmuir equation obtains a homogenous surface) [

39]. The maximum adsorption capacity (

q_{m}) obtained from the Langmuir isotherm was 32.26 mg/g. The magnitude of the exponent

n shows the favorability of adsorption;

n > 1 and

n < 1 represent good and poor adsorption characteristics, respectively. Freundlich and Langmuir adsorption isotherms generally indicate the surface heterogeneity and homogeneity, respectively [

42,

43].

The Brunauer-Emmett-Teller (BET) model was used to determine the monolayer capacity per unit bulk mass of the adsorbed molecules. The BET model follows from Equation (9):

where

C_{e} is the equilibrium concentration (mg/L),

C_{s} is the adsorbate monolayer saturation concentration (mg/L), and

C_{BET} is the BET adsorption isotherm related to the surface interaction energy (L/mg). The

q_{e} is adsorption capacity in equlibrium condition.

One of the modified Langmuir equations is the Toth isotherm, which caused a reduction of error in the experimental and predicted data. This model is the most appropriate method in adsorbate concentration and explains that a system of heterogeneous adsorption satisfies both low and high boundaries. The Toth isotherm model is stated below:

where

K_{e},

K_{L} and

n are Toth isotherm constants (mg·g

^{−1}). The parameter

n is a heterogeneity characteristic; when

n = 1, this equation converts to the Langmuir isotherm equation and, on the other hand, the system is heterogeneous if it deviates further away from unity (1). In this model, the parameters can be examined by a nonlinear curve-fitting method with a sigma plot software (version 14). In addition, for several adsorption multilayer systems, a heterogeneous isotherm of the Toth model has been applied.

Notwithstanding the accessibility of many isotherm models, the Dubinin-Astakhov (DA) model is the most suitable candidate for volume filling of micropores and the adsorption into mesopores and macropores. Moreover, the DA model accounts for the energetic surface heterogeneity of the adsorbents. Furthermore, in order to allow a basic increase in the activation energy in the finest micropores, the DA model uses the Weibull distribution to describe the degree of adsorption with regard to the energy of adsorption. The DA model was introduced in the following form:

where

n_{0} is the maximum capacity per unit aggregate mass of the adsorbed probe molecules at the end of adsorption and

B and

k are fitting parameters. The

C_{o} is initial concentration.

Figure 15 shows the data fitting for these isotherms. As can be seen, the equilibrium data fitted well with these isotherms. However, the Toth model showed a better fitting in comparison with the BET and DA models.