## 3 Results and Discussion
### Adsorption Isotherm Models
#### 3.3.3 Jovanovic-Freundlich Model Isotherm for WPET-SiAMPy Composite Nanofibres  
The Jovanovic model isotherm adopts the same theories as the Langmuir model isotherms and further considers the mechanism between the adsorbate and adsorbent contact [52]. For heterogeneous surfaces, the Jovanovic model extends a semiempirical model named Jovanovic-Freundlich model. The Jovanovic-Freundlich model isotherm was derived from a differential relationship relating the surface coverage and the bulk concentration of the adsorbate for a single component adsorption. The Jovanovic-Freundlich model is reduced to the Jovanovic model when the surface is homogenous, and at high concentrations, it is reduced to a monolayer; however, it does not obey [PERSON]'s law [53]. The Jovanovic-Freundlich model isotherm is expressed as follows:  
\[q_{e}=q_{IF}\left(1-e^{(-K_{IF}c_{e}^{n_{IF}})}\right) \tag{8}\]  
where \(q_{IF}\) is the maximum monolayer coverage capacity, \(K_{IF}\) is the equilibrium constant, and \(n_{IF}\) is the Jovanovic-Freundlich model isotherm exponent exponent.  
Figure 11 shows the Jovanovic-Freundlich isotherm calculated for WPET-SiAMPy nanofibres. The three parameters of the semi-empirical model isotherm were applied to evaluate the Jovanovic-Freundlich model isotherm fit for Cu\({}^{2+}\) adsorption. The Jovanovic-Freundlich model isotherm exponent \(n_{IF}=0.47\) indicated that the Cu\({}^{2+}\) ion binding adsorption occurred at a heterogeneous surface as the value of parameter \(n_{IF}\) is far greater than unity 1, which means the Jovanovic-Freundlich equation was reduced to the Freundlich isotherm. The correlation coefficient was R\({}^{2}=0.8445\), which indicated a favourable adsorption process. [PERSON] et al. [54] reported the derivation and application of a Jovanovic-Freundlich isotherm model for single component adsorption on heterogeneous surfaces.  
The calculated results of all the isotherms modelled for WPET-SiAMPy nanofibres are presented in Table 2.  
The Langmuir, Freundlich, and Jovanovic-Freundlich isotherm models were applied to describe the Cu\({}^{2+}\) ions' adsorption mechanism on the WPET-SiAMPy nanofibres surface using a range of Cu\({}^{2+}\) aqueous concentrations. The experimental data obtained for adsorption of Cu\({}^{2+}\) using WPET-SiAMPy nanofibres were compared to the theoretical data obtained from the [PERSON], Freundlich, and Jovanovic-Freundlich isotherm models, and determined that the three isotherms showed the correlation coefficient (R\({}^{2}\)) following this sequence: Freundlich > Langmuir > Jovanovic-Freundlich [55]. Thus, the Freundlich isotherm model with the highest correlation coefficient R\({}^{2}=0.9952\) was the best fit for Cu\({}^{2+}\) adsorption experimental outcomes. The [PERSON] isotherm model had a good agreement with the experimental data obtained, based on the reduced chi-square X\({}^{2}\) that was far less than unity, good correlation coefficient R\({}^{2}\), and the close approximation of the theoretical maximum uptake of adsorbate \(q_{IF}\) to the experimental adsorption uptake of adsorbate \(q_{e}\). Thus, among the three important compatible applicable isotherm models about adsorption, the [PERSON] isotherm model had the best fit to experimental outcomes of Cu\({}^{2+}\) ions adsorption from aqueous solution using the WPET-SiAMPy nanofibres. This indicated that the WPET-SiAMPy nanofibres' adsorption sites for Cu\({}^{2+}\) ions were uneven and not restricted to a particular adsorption site. These results proposed that more than one type of WPET-SiAMPy active site took part in Cu\({}^{2+}\) ion adsorption [56].