Physical chemistry is such a broad discipline that the topics we expect average students to complete in two semesters usually exceed their ability for meaningful learning. Consequently, the number and kind of topics and the efficiency with which students can learn them are important concerns. What topics are essential and what can we do to provide efficient and effective access to those topics? How do we accommodate the fact that students come to upper-division chemistry courses with a variety of nonuniformly distributed skills, a bit of calculus, and some physics studied one or more years before physical chemistry?

The critical balance between depth and breadth of learning in courses and curricula may be achieved through appropriate use of technology and especially through the use of symbolic mathematics software. Software programs such as Mathcad, Mathematica, and Maple, however, have learning curves that diminish their effectiveness for novices. There are several ways to address the learning curve conundrum. First, basic instruction in the software provided during laboratory sessions should be followed by requiring laboratory reports that use the software. Second, one should assign weekly homework that requires the software and builds student skills within the discipline and with the software. Third, a complementary method, supported by this column, is to provide students with Mathcad worksheets or templates that focus on one set of related concepts and incorporate a variety of features of the software that they are to use to learn chemistry.

In this column we focus on two significant topics for young chemists. The first is curve-fitting and the statistical analysis of the fitting parameters. The second is the analysis of the rotation/vibration spectrum of a diatomic molecule, HCl.

A broad spectrum of Mathcad documents exists for teaching chemistry. One collection of 50 documents can be found at http://www.monmouth.edu/~tzielins/mathcad/Lists/index.htm. Another collection of peer-reviewed documents is developing through this column at the JCE Internet Web site, http://jchemed.chem.wisc.edu/JCEWWW/Features/ McadInChem/index.html. With this column we add three peer-reviewed and tested Mathcad documents to the JCE site.

In Linear Least-Squares Regression, Sidney H. Young and Andrzej Wierzbicki demonstrate various implicit and explicit methods for determining the slope and intercept of the regression line for experimental data. The document shows how to determine the standard deviation for the slope, the intercept, and the standard deviation of the overall fit. Students are next given the opportunity to examine the confidence level for the fit through the Student's t-test. Examination of the residuals of the fit leads students to explore the possibility of rejecting points in a set of data. The document concludes with a discussion of and practice with adding a quadratic term to create a polynomial fit to a set of data and how to determine if the quadratic term is statistically significant. There is full documentation of the various steps used throughout the exposition of the statistical concepts. Although the statistical methods presented in this worksheet are generally accessible to average physical chemistry students, an instructor would be needed to explain the finer points of the matrix methods used in some sections of the worksheet. The worksheet is accompanied by a set of data for students to use to practice the techniques presented. It would be worthwhile for students to spend one or two laboratory periods learning to use the concepts presented and then to apply them to experimental data they have collected for themselves. Any linear or linearizable data set would be appropriate for use with this Mathcad worksheet. Alternatively, instructors may select sections of the document suited to the skill level of their students and the laboratory tasks at hand.

In a second Mathcad document, Non-Linear Least-Squares Regression, Young and Wierzbicki introduce the basic concepts of nonlinear curve-fitting and develop the techniques needed to fit a variety of mathematical functions to experimental data. This approach is especially important when mathematical models for chemical processes cannot be linearized. In Mathcad the Levenberg-Marquardt algorithm is used to determine the best fitting parameters for a particular mathematical model. As in linear least-squares, the goal of the fitting process is to find the values for the fitting parameters that minimize the sum of the squares of the deviations between the data and the mathematical model. Students are asked to determine the fitting parameters, use the Hessian matrix to compute the standard deviation of the fitting parameters, test for the significance of the parameters using Student's t-test, use residual analysis to test for data points to remove, and repeat the calculations for another set of data. The nonlinear least-squares procedure follows closely on the pattern set up for linear least-squares by the same authors (see above). If students master the linear least-squares worksheet content they will be able to master the nonlinear least-squares technique (see also refs 1, 2).

In the third document, The Analysis of the Vibrational Spectrum of a Linear Molecule by Richard Schwenz, William Polik, and Sidney Young, the authors build on the concepts presented in the curve fitting worksheets described above. This vibrational analysis document, which supports a classic experiment performed in the physical chemistry laboratory, shows how a Mathcad worksheet can increase the efficiency by which a set of complicated manipulations for data reduction can be made more accessible for students. The increase in efficiency frees up time for students to develop a fuller understanding of the physical chemistry concepts important to the interpretation of spectra and understanding of bond vibrations in general.

The analysis of the vibration/rotation spectrum for a linear molecule worksheet builds on the rich literature for this topic (3). Before analyzing their own spectral data, students practice and learn the concepts and methods of the HCl spectral analysis by using the fundamental and first harmonic vibrational frequencies provided by the authors. This approach has a fundamental pedagogical advantage. Most explanations in laboratory texts are very concise and lack mathematical details required by average students. This Mathcad worksheet acts as a tutor; it guides students through the essential concepts for data reduction and lets them focus on learning important spectroscopic concepts. The Mathcad worksheet is amply annotated. Students who have moderate skill with the software and have learned about regression analysis from the curve-fitting worksheets described in this column will be able to complete and understand their analysis of the IR spectrum of HCl.

The three Mathcad worksheets described here stretch the physical chemistry curriculum by presenting important topics in forms that students can use with only moderate Mathcad skills. The documents facilitate learning by giving students opportunities to interact with the material in meaningful ways in addition to using the documents as sources of techniques for building their own data-reduction worksheets. However, working through these Mathcad worksheets is not a trivial task for the average student. Support needs to be provided by the instructor to ease students through more advanced mathematical and Mathcad processes. These worksheets raise the question of how much we can ask diligent students to do in one course and how much time they need to spend to master the essential concepts of that course.

The Mathcad documents and associated PDF versions are available at the JCE Internet WWW site. The Mathcad documents require Mathcad version 6.0 or higher and the PDF files require Adobe Acrobat. Every effort has been made to make the documents fully compatible across the various Mathcad versions. Users may need to refer to Mathcad manuals for functions that vary with the Mathcad version number.

Literature Cited

1. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969.
2. Zielinski, T. J.; Allendoerfer, R. D. J. Chem. Educ. 1997, 74, 1001.
3. Schwenz, R. W.; Polik, W. F. J. Chem. Educ. 1999, 76, 1302.