On the use of Linearized Langmuir Equations

Authors: Bolster, Carl; HORNBERGER, GEORGE - UNIVERSITY OF VIRGINIA

Submitted to: Soil Science Society of America Journal
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 5/25/2007
Publication Date: 11/1/2007
Citation: Bolster, C.H., Hornberger, G.M. 2007. On the use of Linearized Langmuir Equations. Soil Science Society of America Journal. Vol.71 Issue 6 P 1796-1806

Interpretive Summary: The transport behavior of environmentally significant reactive solutes such as phosphorus and heavy metals is controlled in large part by the sorption behavior of these solutes to soil surfaces. To obtain soil sorption parameters requires that a sorption model be fit to sorption isotherm data. One of the most commonly used models for describing solute sorption to soils is the Langmuir model. The Langmuir model can be fit directly to sorption data through the use of nonlinear regression or a linearized version of the Langmuir equation can be used. Because linear regression is convenient, requires little understanding of the data fitting process, and is easily done in spreadsheets such as Microsoft Excel, this method is commonly used for obtaining Langmuir sorption parameters. A limitation to this approach, however, is that the transformation of data required for linearization can result in differences in fitted parameter values between linear and nonlinear versions of the Langmuir model. In addition, the ease of using linear regression may discourage the critical evaluation of the model fits to the data. In this study we set out to further our understanding of the limitations of using linearized Langmuir equations. Such understanding is particularly important in phosphorus (P) sorption studies where linearized Langmuir equations are commonly used to obtain important sorption parameters which are used in land management decisions. Our results demonstrate that the reliance on linearized Langmuir equations potentially limits the ability to model sorption data accurately. Therefore, to encourage the testing of more sophisticated nonlinear sorption models we make available an accurate and easy-to-use Microsoft Excel spreadsheet capable of performing nonlinear regression. Results of this study will allow researchers to make more informed decisions when applying the Langmuir model to their sorption data.

Technical Abstract: One of the most commonly used models for describing solute sorption to soils is the Langmuir model. Because the Langmuir model is nonlinear, fitting the model to sorption data requires that the model be solved iteratively using an optimization program. To avoid the use of optimization programs, a linearized version of the Langmuir model is often used so that model parameters can be obtained by linear regression. Although the linear and nonlinear Langmuir equations are mathematically equivalent, there are several limitations to using linearized Langmuir equations. First, the transformation of data required for linearization can lead to statistical differences between the linear and nonlinear equations. Second, the ease of using linearized Langmuir equations may discourage critical evaluation of model fits. And finally, relying solely on linearized equations for describing sorption data prevents the testing of more sophisticated models which cannot be linearized. We examined the limitations of using linearized Langmuir equations by fitting phosphorus (P) sorption data collected on eight different soils with four linearized versions of the Langmuir equation and comparing goodness-of-fit measures and fitted parameter values with those obtained with the nonlinear Langmuir equation. We then fit the sorption data with two modified versions of the Langmuir model and assessed whether the fits were statistically superior to the original Langmuir equation. Although the nonlinear Langmuir equation did provide slightly different parameter estimates than did the four linearized versions of the Langmuir equation, these differences were minor when compared to the improvement in fit obtained with the modified Langmuir equations. Our results demonstrate that the use of linearized Langmuir equations needlessly limits the ability to model sorption data with good accuracy. To encourage the testing of additional nonlinear sorption models, we make available an easily used Microsoft® ExcelTM spreadsheet (http://ars.usda.gov/msa/awmru/bolster/Sorption_spreadsheets) capable of generating best-fit parameters, standard errors of the parameters, confidence intervals for the parameters, correlations between fitted parameters, and goodness-of-fit measures. The results of our study should promote more critical evaluation of model fits to sorption data and encourage the testing of more sophisticated sorption models.