## 3 Results and Discussion
### Adsorption Isotherm Models  
The adsorption isotherm models of Langmuir and [PERSON] were used to illustrate the mechanism of Cu\({}^{2+}\) adsorption using the AgNPs.  
The [PERSON] and [PERSON] adsorption isotherm models are expressed as follows [32]:  
\[\frac{C_{e}}{q_{e}}=\frac{1}{K_{L}q_{m}}+\frac{C_{e}}{q_{m}}, \tag{7}\]  
\[R_{L}=\frac{1}{1+K_{L}C_{0}}, \tag{8}\]  
\[log\ q_{e}=log\ K_{F}+\frac{1}{n}\ log\ C_{e}, \tag{9}\]  
where \(C_{e}\) (mg/L) is the concentration of the adsorbate at equilibrium, \(q_{e}\) is the capacity of adsorption at equilibrium, \(q_{m}\) is the maximum capacity of adsorption, \(K_{L}\) is the constant of Langmuir adsorption at equilibrium, and n is the coefficient of Freundlich at equilibrium.  
The Langmuir isotherm hypothesis is valid for a monolayer forming on the adsorbent surface, whereas the Freundlich isotherm is an experimental equation that is applicable to a multilayer adsorption process.  
Figure 8, shows that there is a linear relationship between the \(C_{e}\)/\(q_{e}\) and \(C_{e}\) and between the log \(q_{e}\) and log \(C_{e}\) for the Cu\({}^{2+}\) adsorption by the AgNPs. The plots slopes and the correlation coefficients (\(R^{2}\)) are recorded in Table 2. The \(R^{2}\) values indicated that the Langmuir imore suitable for the experimental than the Freundlich; moreover, the theoretical maximum adsorption capacity of a monolayer coverage (\(q_{m}\)) was shown to be higher than the experimental capacity [33].  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Kinetic Models**} & \multicolumn{2}{c}{**Adsorbent**} \\  & **10 mg/L** & **50 mg/L** \\ \hline \(q_{e,\text{exp}}\) mg/g & **2.278** & **9.225** \\ \hline Pseudo-First-Order & & \\ \(q_{e,\text{cal}}\) mg/g & 1.401 & 6.391 \\ \(K_{1}\), min\({}^{-1}\) & 0.031 & 0.031 \\ \(R^{2}\) & 0.996 & 0.945 \\ \hline \multicolumn{3}{c}{**Pseudo-Second-Order**} \\ \(q_{e,\text{cal}}\) mg/g & **2.470** & **10.823** \\ \(K_{2}\), g/mg min & **0.033** & **0.004** \\ \(R^{2}\) & **0.999** & **0.984** \\ \hline \multicolumn{3}{c}{**Intraparticle Diffusion**} \\ \hline \multicolumn{3}{c}{**Step 1**} \\ \(K_{jpd}\), mg/g.min\({}^{1/2}\) & **0.149** & **1.419** \\ \(C\), mg/g & **0.898** & **2.437** \\ \(R^{2}\) & **1** & **1** \\ \hline \multicolumn{3}{c}{**Step 2**} \\ \(K_{jpd}\), mg/g.min\({}^{1/2}\) & **0.024** & **0.074** \\ \(C\), mg/g & **1.979** & **8.259** \\ \(R^{2}\) & **0.932** & **0.975** \\ \hline \multicolumn{3}{c}{**Liquid Film Diffusion**} \\ \(K_{jpd}\), min\({}^{-1}\) & **0.031** & **0.031** \\ \(\bar{C}\), mg/g & **0.487** & **0.368** \\ \(R^{2}\) & **0.996** & **0.945** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of four kinetic models.  
The physicochemical constant of dimensional was applied as a factor of separation (\(R_{L}\)) to determine whether the adsorption process was favorable, which is expressed in Equation (8) [31]. \(R_{L}\), the separation factor for the adsorption of Cu\({}^{2+}\) by the AgNPs, was 0.254 at 298 K and 0.426 at 323 K, indicating a favorable adsorption.  
Generally, \(R_{L}\) indicates that the isothermal type is favorable (0 < \(R_{L}\) < 1), linear (\(R_{L}\) = 1), unfavorable (\(R_{L}\) > 1), or irreversible (\(R_{L}\) = 0).  
The values of n to assess the adsorption process suitability were 1.883 at 298 K and 1.807 at 323 K, indicating that the Cu\({}^{2+}\) ions adsorption by the AgNPs is favorable [30].  
Generally, the value of n is the adsorption intensity: n = 1 indicates the homogeneity of adsorption sites and lack of interaction between occupied sites; when n is between 1 and 10 it indicates a favorable adsorption, and when n < 1 a cooperative adsorption is indicated. From the above results, it was found that the Cu\({}^{2+}\) ions adsorption by the AgNPs is a favorable process and that the Cu\({}^{2+}\) ions are adsorbed on the AgNPs surface as monolayers and multilayers [15].