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On the Use of Linearized Langmuir Equations

^a USDA-ARS, 230 Bennett Ln., Bowling Green, KY 42104
^b Dep. of Environmental Sciences, Univ. of Virginia, Charlottesville, VA 22903

View larger version (6K): [in this window] [in a new window]   Fig. 1. Relationship between number of data points and reduction in the sum of squared errors (SSE) required to obtain a 95% probability that a three- or four-parameter model is superior to a two-parameter model using Akaike's Information Criterion (AIC). For example, for seven data points, nearly a four order-of-magnitude reduction in SSE is required to obtain a 95% probability that the fit provided by a four-parameter model is superior to the fit provided by a two-parameter model. On the other hand, for 10 data points, the required reduction in SSE is less than one order of magnitude.

View larger version (15K): [in this window] [in a new window]   Fig. 2. Example of Excel spreadsheet for fitting nonlinear sorption equations to sorption isotherm data.

View larger version (11K): [in this window] [in a new window]   Fig. 3. Comparisons of fitted parameter values and statistics (K is the Langmuir binding-strength coefficient, SSE is the sum of squared errors, Smax is the maximum sorption capacity of the soil, and β is the exponent in the Langmuir–Freundlich model) between SAS and Excel obtained by fitting the Langmuir, Langmuir–Freundlich, and two-surface Langmuir models to sorption data using unweighted nonlinear least squares regression. Parameter values and statistics all fall on the 1:1 line, indicating that the Excel spreadsheet yields nearly identical values as SAS.

View larger version (11K): [in this window] [in a new window]   Fig. 4. Comparisons of fitted parameter values and statistics (K is the Langmuir binding-strength coefficient, SSE is the sum of squared errors, Smax is the maximum sorption capacity of the soil, and β is the exponent in the Langmuir–Freundlich model) between SAS and Excel obtained by fitting the Langmuir, Langmuir–Freundlich, and two-surface Langmuir models to sorption data using weighted (w = 1/S2, where w is the weighting factor and S is the measured value) nonlinear least squares regression. Parameter values and statistics all fall on the 1:1 line, indicating that the Excel spreadsheet yields nearly identical values as SAS.

View larger version (15K): [in this window] [in a new window]   Fig. 5. Comparisons of fitted (A) maximum sorption capacity of the soil (Smax) and (B) Langmuir binding-strength coefficient (K) values between the four linearizations of the Langmuir equation and the nonlinear Langmuir equation. The bars represent the 95% confidence intervals of the fitted parameter values obtained with the nonlinear Langmuir equation.

View larger version (9K): [in this window] [in a new window]   Fig. 6. Measured sorption data and model fits for (A) Collins, (B) Loring, and (C) Pembroke soils.

View larger version (8K): [in this window] [in a new window]   Fig. 7. Model fits to sorption data measured for the Hartsells soil obtained with (A) Linearization I (Table 1) and (B) the nonlinear equation. Also shown in (B) is the model fit obtained with Linearization I but transformed to sorbed concentration.