Comparisons of linear and nonlinear Langmuir and Freundlich curve-fit in the study of Cu, Cd, and Pb adsorption on Taiwan soils

In the Langmuir adsorption equation, q = MbC/(1+bC), the b parameter can be identified as the reciprocal of the concentration, C-1/2, at which the adsorbent is half-saturated with the adsorbate. If the concentration, C, is scaled in the unit of C-1/2, and replaced the C', where C' = C/C-1/2, the universal dimensionless Langmuir equation, theta = C'/(1+C'), is obtained, Arbitrary points chosen on segments of the normal Langmuir plot can be fitted to different Freundlich equations with statistical significance. This indicates that the Freundlich equation can be applied to represent a selected range of the adsorption data that also fit the Langmuir equation. Linear and nonlinear least squares methods were applied to fit experimental data of adsorption of a metal ion in the presence of another metal ion, on three Taiwan soils, to Langmuir and Freundlich equations. The goodness-of-fit of the model to the experimental data was compared with the magnitude of the residual root mean square error (RMSE) of the original nonlinear forms of both adsorption isotherms. Results indicate that simple conclusions, based on the R-2 values obtained by the usual linear least squares method applied to the linearly transformed equations, may be in error. Even when the metal ion adsorption on soils appeared to be better represented by the Freundlich equation,judging from the size of the R-2 value, than by the Langmuir equation, there are cases in which the Langmuir equation could better represent the experimental data based on the size of RMSE value,These were examples of experiments conducted in a limited concentration range, Increasing the range of concentration for the adsorption experiments may eventually turn the Freundlich-type adsorption isotherms into the Langmuir type if no complication arises in the more concentrated solutions.