On the Use of Linearized Langmuir Equations

Soil Sci. Soc. Am. J. 71:1796–1806 doi:10.2136/sssaj2006.0304 Received 30 Aug. 2006. *Corresponding author (carl.bolster@ars.usda.gov). © Soil Science Society of America 677 S. Segoe Rd. Madison WI 53711 USA All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher. The transport behavior of environmentally signifi cant reactive solutes such as P and heavy metals is controlled in large part by the sorption behavior of these solutes to soil surfaces. Sorption behavior to soils is often determined by measuring sorption isotherms, where a known mass of soil is equilibrated with a solution of known concentration of the solute of interest. After equilibration, the concentration remaining in solution is measured and used to calculate the concentration sorbed to the soil (Nair et al., 1984). A sorption model is fi t to the data to obtain sorption parameters for the soil. These sorption parameters are used to estimate parameters such as the sorption capacity of the soil or retardation coeffi cients to be used in transport modeling. The accuracy of the model parameters will depend on whether the appropriate conceptual model was chosen, whether the experimental conditions were representative of environmental conditions, and whether an appropriate parameter estimation method was used. A commonly used model for describing sorption behavior is the Langmuir model (Altin et al., 1998; Kumar and Sivanesan, 2005; Kleinman and Sharpley, 2002; Tsai and Juang, 2000; Wang and Harrell, 2005). Because the Langmuir model is nonlinear, fi tting this model to measured data requires a “trial and error” approach. That is, values of the parameters are inserted into the model, the sorbed concentrations are calculated with the model, the model-calculated values are then compared with the observed data, the model parameters are adjusted, and the process is repeated until the best agreement between modeled and observed data is achieved. Alternatively, a linearized version of the Langmuir equation—at least four different versions exist (Table 1)—can be used so that the model parameters can be obtained directly by solving the normal equations (i.e., by linear regression). Because linear regression is convenient, requires little understanding of the data-fi tting process, and is easily done in spreadsheets such as Microsoft Excel, this method is commonly used for obtaining Langmuir sorption parameters (Borling et al., 2001; D’Angelo et al., 2003; Fang et al., 2002; Kleinman and Sharpley, 2002; Sharpley, 1995; Siddique and Robinson, 2003; Xu et al., 2006; Zhang et al., 2005). A limitation to this approach, however, is that the transformation of data required for linearization can result in modifi cations of error structure, introduction of error into the independent variable, and alteration of the weight placed on each data point (Dowd and Riggs, 1965; Harter, 1984), often leading to differences in fi tted parameter values between linear and nonlinear versions of the Langmuir model (Altin et al., 1998; Kumar and Sivanesan, 2005; Schulthess and Dey, 1996; Tsai and Juang, 2000). Although it is commonly assumed that linearized versions of the Langmuir model provide poorer fi ts and less accurate parameter estimates than the nonlinear equation (Harrison and Katti, 1990; Kumar and Sivanesan, 2005; Kinniburgh, 1986), the most accurate Langmuir equation will depend on the error structure of the data because a major assumption in regression analyses is that the variance of the errors remains constant. Therefore, if a Carl H. Bolster* USDA-ARS 230 Bennett Ln. Bowling Green, KY 42104