# Adsorption of Hexavalent Chromium Ions Using Pine Sawdust Cellulose Fibres
## 3 Results and Discussion
### Error Analysis for Isotherm  
Three distinct linear error functions are utilised to assess the goodness of fit of the isotherms, and these functions are considered an effective method for analysing experimental data generated from the adsorption process. Sum of the Squares of the Errors (SSE). The sum of the squares of the errors (SSE) is similar to the Sum of the Absolute Errors Function (SAE), which is the most-used error function (Equation (7)) [63]:  
\[\rm SSE=\sum_{i=1}^{n}(q_{e}-q_{m})^{2} \tag{7}\]  
Residual root-mean-square error (RMSE). RMSE is also a widely used error evaluation function, and the mathematical form is  
\[\rm RMSE=\sqrt{\sum_{i=1}^{n}(q_{e}-q_{m})^{2}} \tag{8}\]  
This error evaluation function is used if the deviation between the experimental and predicted values is significant. Additionally, RMSE is commonly used to prevent models that generate occasional significant errors and adhere to the normal distribution, which is the foundation for fitting ordinary least squares regression models [64]. Chi-square test, \(\chi^{2}\) can be used to confirm the best-fit isotherm for the adsorption system (Equation (9)); if the results from the model are similar to the experimental results, \(\chi^{2}\) is a small number; if they are different, \(\chi^{2}\) is a large number [65]. \[\chi^{2}=\sum\frac{\left(q_{e}-q_{m}\right)^{2}}{q_{m}} \tag{9}\]  
where \(q_{m}\) is the equilibrium capacity obtained by calculating from the model (mg/g), and \(q_{e}\) is the equilibrium capacity (mg/g) experimental data. The parameter \(\rm k_{F}\) in the Freundlich equation represents the adsorption capacity of the adsorbent. In both the linear and nonlinear models, the \(\rm k_{F}\) value was approximately 10, indicating a moderate adsorption capacity. The high \(\rm k_{F}\) values implied a significant uptake of chromium ions onto the absorbent surface [62]. Table 6 demonstrates that the nonlinear Langmuir model was better than the linear Langmuir model in various aspects. It had a significantly lower sum of squares error (SSE) of 0.1158 compared to 1.84 \(\times\) 10\({}^{-2}\), indicating a better fit with reduced discrepancies between the observed and predicted values. The root-mean-square error (RMSE) was also lower in the nonlinear model (0.1079) compared to the linear model (1.46 \(\times\) 10\({}^{-2}\)), suggesting more minor average deviations. The \(\chi^{2}\) value for the nonlinear Langmuir model was much lower (3.14 \(\times\) 10\({}^{-7}\)) than that of the linear model (8.40 \(\times\) 10\({}^{-2}\)), indicating a superior fit. Additionally, the nonlinear model had a higher R\({}^{2}\) value of 0.9992 than the linear model's R\({}^{2}\) value of 0.9985, explaining a more significant proportion of the total variation in the data. Moreover, the Sum of Normalised Errors (SNE) was substantially smaller for the nonlinear model (6.976 \(\times\) 10\({}^{-4}\)) than the linear model (2.78 \(\times\) 10\({}^{-2}\)), further confirming it was a better fit. The nonlinear Freundlich model (Figure 16b) had a higher SSE of 5.44 \(\times\) 10\({}^{-3}\) than the linear Freundlich model's SSE of 2.82 \(\times\) 10\({}^{-4}\). This indicated that the linear Freundlich model provides a better fit due to its smaller sum of squares error, implying reduced differences between the observed and predicted values. Additionally, the linear Freundlich model had a lower RMSE of 7.13 \(\times\) 10\({}^{-3}\) than the nonlinear Freundlich model's RMSE of 3.46 \(\times\) 10\({}^{-2}\), suggesting minor average deviations between the observed and predicted values. Moreover, the \(\chi^{2}\) value for the linear Freundlich model (4.91 \(\times\) 10\({}^{-5}\)) was significantly lower than the nonlinear Freundlich model's \(\chi^{2}\) value (6.20 \(\times\) 10\({}^{-3}\)), indicating a better fit for the linear model.