1. Statistical Analysis of Fitness of Various Isotherms to Experimental Sorption Data
Hsien-Chun Lina, Ming-Yao Chena, Hua-Yen Chia, Yuh-Shan Hob#* and Wen-Ta Chiua
aTaipei Medical University – Wan Fang Hospital,
bSchool of Public Health, Taipei Medical University
Introduction
The most common isotherms applied in solid/liquid system are the theoretical equilibrium isotherm, Langmuir [1], the best known and most often used isotherm for the sorption of a solute from a liquid solution; the Freundlich [2], the earliest known relationship describing the adsorption equation and the Redlich-Peterson [3], the earlier presented, containing three parameters isotherm. In this study, a comparison of linear regression and Chi-square analysis of three isotherms, Langmuir, Freundlich and Redlich-Peterson, have been applied to the experiment of cadmium sorption on tree fern.
Materials and Methods
A volume of 50 ml of Cd(II) solution with a concentration in the properly range was placed in a 125 ml conical flask. An accurately weighed sugar cane sample 0.2 g was then added to the solution. A series of such conical flasks was then shaken at a constant speed of 100 rpm in a shaking water bath with temperatures of 25°C. After shaking the flasks for 5 h, the tree fern was separated by filtration through a Nylon membrane filter (0.45 µm). These were analysed by an inductively coupled plasma atomic emission spectroscopy analyser (ICP-AES) for the concentration of cadmium.
Results and Discussion
An error function is required to evaluate the fitness of the equation to the experimental data obtained from the optimization process employed. In this study, the linear coefficient of determination and non-linear Chi-Square determination was applied. The linear coefficient of determination, r2, found from evaluation of data by linear model, was calculated with the aid of the equation:
Table 2. Comparison of linear regression r, r2, and non-linear Chi-square test analysis, 2.
where Sxx the sum of squares of X; Syy the sum of squares of Y; Sxy the sum of squares of X and Y
The Chi-square test: The equivalent mathematical statement is
Figure 1. Theoretical isotherms and experimental data
where qe,m equilibrium capacity obtained by calculating from model (mg/g), qe experimental data of equilibrium capacity (mg/g).
Table 2 shows that if just using the linear form of Langmuir-1 for comparison, Redlich-Peterson isotherm was most suitable for the data followed by Langmuir-1 and then Freundlich isotherm. In contrast, if using the linear form of Langmuir-2, Redlich-Peterson isotherm was still most suitable, but was followed by Freundlich and then Langmuir isotherm. Three linear equations had different axial settings individually so that would alter the result of a linear regression and influence the determination process.
2 of the two Langmuir isotherms were similar where the Redlich-Peterson and Freundlich isotherm exhibited identical and lower 2 values than Langmuir and the latter ones were considered to be a better match. The Redlich-Peterson and Freundlich isotherms almost overlapped and seemed to be the best-fitting models for the experiment results from Fig. 1. Consequently, the Redlich-Peterson and Freudlich isotherms were the most suitable models for this sorption system. Unlike the linear analysis, different forms of equation would effect r and r2 significantly and impact the final determination where non-linear Chi-square analysis would be a method of avoiding such errors.
Table 1. Isotherms and Their Linear Forms
Conclusion
In this study, we would like to point out that it is not appropriate to use the correlation coefficient (r) or the coefficient of determination (r2) of linear regression analysis for comparing the best-fitting of Freundlich and both linear Langmuir isotherms. Non-linear Chi-square analysis could be a better method. Freundlich is a special case of Redlich-Peterson isotherm when constants A and B were much bigger than 1. It was clear that both two-parameter Freundlich and three-parameter Redlich-Peterson isotherms were the best-fitting models for the experiment results.
References
Langmuir, I. (1916), J. Am. Chem. Soc., 38,
Freundlich, H.M.F. (1906), Z. Phys. Chem., 57,
Redlich, O. and Peterson, D.L. (1959), J. Phys. Chem., 63, 1024.