# Adsorption of Ni2+ and Cd2+ from Water by Calcium Alginate/Spent Coffee Grounds Composite Beads
## 3 Results and Discussion
### Analysis of Adsorption Isotherms  
In this study, three models were used: Langmuir, Freundlich, and Sips. The Langmuir model is the most commonly used model of adsorption isotherm and it assumes monolayer adsorption (the adsorbed layer is one molecule in thickness). The adsorption can only occur at a finite (fixed) number of definite localized sites that are identical and equivalent. The non-linear Langmuir model is described in Equation (4) [15]:  
\[q_{e}=(q_{max}\times b_{L}\times C_{e})/(1+b_{L}\times C_{e}) \tag{4}\]  
where \(q_{max}\) is the maximum monolayer coverage capacity (mg/g) and \(b_{L}\) is the Langmuir isotherm constant (L/mg) related to the affinity. Freundlich model is applied to multilayer adsorption, with non-uniform distribution of adsorption heat and affinities over the heterogeneous surface. The stronger binding sites are occupied first, until adsorption energy is exponentially decreased upon the completion of adsorption process. The non-linear Freundlich model can be represented by Equation (5) [31]:  
\[q_{e}=K_{f}\times C_{e}\sfrac{1}{n} \tag{5}\]  
where \(K_{f}\) is the adsorption capacity (mg/g)(L/mg)(1/n) and 1/\(n\) is the adsorption intensity, which also indicates the relative distribution of the energy and the heterogeneity of the adsorbate sites, becoming more heterogeneous as its value gets closer to zero. Sips model is a combination of the Langmuir and Freundlich models. The non-linear Sips model is described in Equation (6) [32]:  
\[q_{e}=(q_{max}\times b_{S}\times C_{e}\sfrac{1}{n})/(1+b_{S}\times C_{e}\sfrac {1}{n}) \tag{6}\]  
where \(q_{e}\) is the adsorbed amount at equilibrium (mg/g), \(C_{e}\) is the equilibrium concentration of the adsorbate (mg/L), \(q_{max}\) is the Sips maximum adsorption capacity (mg/g), \(b_{S}\) is the Sips equilibrium  
Figure 9: Amounts Cd\({}^{2+}\) or Ni\({}^{2+}\) absorbed \({}^{2+2+}\)at 24 h for each initial concentration of metal: (a) Percentages of removal at 24 h and (b) Experimental amounts of metal adsorbed at equilibrium (\(q_{e}\)). constant (L/mg), and 1/\(n\) is the Sips model exponent. Values for 1/\(n\) <1 indicate heterogeneous adsorbents, while values closer to 1.0 indicate a material with homogenous binding sites. If 1/\(n\) =1, the Sips model is reduced to the Langmuir equation [33]. Non-linear equations were used for the three models; therefore, all model parameters were evaluated by non-linear regression using an optimization routine of the Microsoft Excel solver tool. The objective was to minimize the sum of squared errors (SSE) in the concentration range used in adsorption experiments. This method seeks to minimize the value of SSE between observed and calculated values of the dependent variable. The value of SSE was calculated from Equation (7) [34]:  
\[\text{SSE}=\sum_{i=1}^{n}\left[\text{q}_{\text{exp}}-\text{q}_{\text{model}} \right]^{2} \tag{7}\]  
where \(\text{q}_{\text{exp}}\) is the experimental data, and \(\text{q}_{\text{model}}\) is the value predicted by the model at the corresponding \(C_{e}\). To compare the applicability of isotherm equations and fitting it to the data, non-linear chi-square test \(\chi^{2}\) was calculated from Equation (8) [33]:  
\[\chi^{2}=\sum_{n=1}^{n}\left[\left(\text{q}_{\text{e.exp.n}}-\text{q}_{\text{ e.model.n}}\right)^{2}/\text{q}_{\text{e.exp.n}}\right] \tag{8}\]  
where \(\text{q}_{\text{e.exp}}\) is the equilibrium sorption capacity found from the batch experiment, \(\text{q}_{\text{e.model}}\) is the prediction from the isotherm model for corresponding to \(C_{e}\), and n is the number of observations. Tables 2 and 3 show the results of the adsorption models for each adsorbent. The larger values of non-linear chi-square tests \(\chi^{2}\) for the Langmuir model of the CA-SCGs beads indicate a more heterogeneous adsorption by CA-SCGs beads than by CA beads (Tables 2 and 3).