This Isn't Real

Environ. Sci. Techno/. 1986, 20, 895-904

General Purpose Adsorption Isotherms
David G. Kinniburgh
Hydrogeology Research Group, British Geological Survey, Wallingford, Oxon OX10 855, U.K.

The fitting of adsorption isotherm equations to experimental data is often an important aspect of data analysis. If the Langmuir and Freundlich isotherms are used, then consideration must be given to theproper weighting of the observations. Preferably nonlinear regression (nonlinear least squares) should be used since this enables these isotherms to be fitted directly and also enables other isotherms to be tested with little extra effort. Isotherms described here which are likely to show a wide range of applicability include the Tbth, modified Dubinin-Radushkevich, and multisite Langmuirisotherms. These can also describe competitive adsorption (binary exchange) reactions and are well suited for heterogeneous exchangers such as soils and sediments. Specific examples discussed are the adsorption of P and K by soils, Na-Cu exchange by montmorillonite, and Zn adsorption by ferrihydrite. Introduction Although the Langmuir and Freundlich isotherms were first introduced about 70 years ago, theystill remain the two most commonly used adsorption isotherm equations. Their success undoubtedly reflects their ability to fit a wide variety of adsorption data quite well, but it may also partly reflect the appealing simplicity of the isotherm equations and the ease with which their adjustable parameters can be estimated. Both isotherm equations can be transformed to a linear form and so theirtwo adjustable parameters are easily estimated either by graphical means or by linear regression. This ease of fitting may have led to the Langmuir and Freundlich isotherms enjoying rather more “success”than they deserve since closer examination of the data often reveals systematic deviations from the fitted isotherms. For many isotherms, including those with three or more adjustable parameters, itis no longer possible to estimate the adjustable parameters by ordinary linear regression or by any reliable graphical means, and so it is necessary to use nonlinear regression. This usually involves the minimization of the residual sums of squares and is often called nonlinear least squares (NLLS). Many NLLS algorithms are now widely available in easy-to-use computer programs, and so the fittingof complicated nonlinear isotherms is no longer difficult. An important advantage of the NLLS approach is that, once set up, it is relatively straightforward to fit a wide range of adsorption isotherm equations with little extra effort. In this paper, it is shown how just a few “general purpose” isotherms can be made to fit a wide variety of adsorption data. Various aspects of the fitting ofadsorption isotherms to experimental data are also discussed. While NLLS methods are preferred even for fitting the Langmuir and Freundlich isotherms, it is likely that linear regression methods will continue to be used. The usual way of fitting the Langmuir and Freundlich isotherms involves fitting one of the transformed forms of the original isotherm equation either by graphical means or byunweighted linear regression. This conveniently ignores the particular distribution of error implied and, as has been repeatedly pointed out for the Langmuir isotherm, cab lead to biased estimates of the isotherm parameters. This is perhaps not so critical when adsorption data are confined to a narrow range of adsorption densities, but it becomes
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increasingly importantas the range increases. Therefore, the question of the proper weighting of the observations when linear regression is used is also briefly discussed. The most promising extensions to the Langmuir and Freundlich isotherms are based on the generalized Langmuir and Generalized exponential isotherms (1). Several special cases of these two isotherms are potentially interesting (Table I), but this...