Least Squares Fitting of Non-Linear Data in the Undergraduate Laboratory

Abstract

Linear least squares fitting of experimental data is a corner stone of data analysis in general chemistry and physical chemistry laboratory programs. Even when nonlinear data are available, students are routinely taught to linearize it and apply linear least squares fitting to extract chemically significant information. This approach is becoming even more strongly entrenched with the availability of spreadsheet programs that are able to linearize data and generate a linear least squares fit with only a few key strokes.There are several reasons why the linearization approach may be inappropriate. First, the power of spreadsheets like Excel and mathematical engines like Mathcad make it possible to extract information directly from minimally processed data through the use of nonlinear curve fitting. These tools permit modeling of the data without introducing the biases implicit in linearizing nonlinear data. Second, the power of the new computer tools available to chemists and chemistry students alleviates the need to depend on techniques that were essential before the advent of the routine use of PCs in the laboratory. Third, students are capable of learning and using nonlinear curve fitting techniques. While younger students may need to use templates, physical chemistry students are ready to program many of these techniques for themselves, especially when using an equation engine, and should be taught to do so.Reduction of kinetic data through nonlinear curve fitting will be the focus of this paper. The methods described can be applied to other types of nonlinear data commonly obtained from the routine use of laboratory instruments. Although the theoretical basis for the approach taken in this paper is not new, the discussion of it is scattered throughout the literature and is not in a form that would be useful to the majority of students in college courses. The summary in this paper should make the theory accessible to junior and senior students and their professors. This is important because as teachers and students implement the statistical features in commercial software, they need to have some experience with the underlying theory of the applications found in these products. DiscussionChemical kinetics is taught to undergraduate chemistry students in a way that often does not correspond to modern practice. In particular, the determination of the order of a chemical reaction is frequently done using the standard linear forms, namely, ln(B 0 /B) or 1/B -1/B 0 is plotted as a function of time for first-and second-order reactions, respectively. Students are advised to make plots corresponding to the expected linearized forms of the kinetic models for the reaction, and then choose the one that appears linear or the one with the best correlation coefficient. The rate constant is then extracted from the slope of the best straight line as determined from visual or linear least squares fitting procedures.A good example of the inadequacy of this technique can be found in the text by Noggle (p 527) (1), where 1st-order,...

Nonlinear Least-Squares Curve Fitting with Microsoft Excel Solver

“…The purpose of this article is to introduce the method of Walsh and Diamond (1) to readers of this Journal, to extend their treatment to weighted least squares, and to add a simple method for estimating uncertainties in the least-square parameters. Other recipes for curve fitting have been presented in numerous previous papers (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16).…”

Which model should be selected to analyze the experimental data among many adsorption kinetic models?

“…MATLAB has a powerful ODE function 47 which are possible to solve by numeral integrated method and the programming are also simple. The kinetic parameters (rate constant, order and so on) were calculated by the nonlinear least squares fitting 48 with "fminsearch" and "fminunc"(MATLAB functions). Figure 1 shows the calculated results of PFO, PSO and MOE in the scaled q/qe-t coordinate.…”

Scientific Data Analysis Toolkit: A Versatile Add-in to Microsoft Excel for Windows

“…As access to computing resources increased, least-squares analysis of data became more widely used and was soon the normal expectation. Many articles published in this Journal describe the application of the least-squares method of data analysis and the importance of the use of such rigorous approaches. − For example, in a seminal publication, de Levie presented the case for using weighted least-squares methods and emphasized the ways and reasons for doing so with data rendered into a linear form via the appropriate transformation of a nonlinear model …”