Estimation of Langmuir Constants using Linear and Nonlinear

Abstract

A nonlinear least squares (NLLS) regression analysis of the Langmuir equation is described here based on minimizing the sum of the normal distance of the data to the isotherm. This regression can yield different Langmuir constants when compared with linear regression methods and another NLLS method — one based on minimizing the sum of the squares of the vertical distance of the data to the isotherm. The two NLLS regressions can also result in two different conclusions (or suggestions) about the physicochemical characteristics of the adsorption phenomena. There is no fundamental, mathematical requirement that the nonlinear regression be based on a vertical minimum save that it is easier to evaluate. More importantly, the vertical NLLS regression is strongly biased toward fitting the low-concentration data; this is remedied by using the normal NLLS regression. None of the regressions are endorsed per se since they should all agree if the isotherm is Langmuirian. The normal NLLS regression method is not sensitive to the goodness-of-fit criteria and, therefore, is considered to be robust. The criteria for choosing a regression method should consider both its sensitivity to data error plus its sensitivity to non-ideality. A deviation in the data is not necessarily due to random measurement errors only (which are often easy to identify when the data are numerous and duplicated), but may also be due to the presence of a partially non-Langmuir adsorption phenomenon. In the latter case, the Langmuir constants usually remain useful, but they must be used cautiously.