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JCE Online: Mathcad in the Chemistry Curriculum
edited by
Theresa Julia Zielinski Monmouth University West Long Beach, NJ 07764
Interactive Fourier Transform Activities In our vigorous teaching of concepts and skills to students, we may cover (hide) more than we uncover, obscuring significant relationships between mathematical models and their associated chemical concepts with excessive mathematical derivations. To set the record straight, I find that mathematical treatments of physical phenomena are beautiful and elegant. Students should know from where the equations and simplifications leading to them arise. They should know the limits of the equations in order to use them properly. However, this can be the Siren’s song. For example, the mathematical representation of the Fourier transform and its significance as presented in most texts are too brief to convey understanding to the typical undergraduate student. Furthermore, a few hand calculations would not permit deeper probing of the method and its intimate link to spectroscopy. The Fourier transform is a very good example of how symbolic equation software can help to uncover the science by making the mathematical manipulations easier and the mathematical concepts more accessible. In July I had the pleasure of attending the 2nd Physical Chemistry Using Mathcad workshop organized by Sidney Young (University of South Alabama), Jeffry Madura (Duquesne University), and Andrzej Wierzbicki (University of South Alabama) at the University of South Alabama. At the workshop thirteen participants, together with the organizers and myself, gathered to learn how to use Mathcad to construct student-centered applications for undergraduate chemistry courses. Our major interest was to learn skills and share ideas about incorporating modern numerical methods into the undergraduate curriculum. I think all participants agreed that to add more numerical methods would require thoughtful selection of what to include and careful planning to deal with the time constraints of a semester or quarter course. At the workshop Peter Atkins (Lincoln College, Oxford), author of one of the most widely used physical chemistry texts, gave a keynote address. He clearly described the role of mathematics in chemistry, characterizing mathematics as the art of rendering the qualitative quantitative. This he counterpoised with the idea that the quality of teaching is not measured by the quantity of material delivered, but rather by the success of the teacher in developing perceptive, adaptable minds. The rendering role of mathematics in physical chemistry is a stumbling block for many students and a challenge for their teachers. How can we get students to do the mathematics required for them to explore the models used by chemists to describe physical phenomena? Complementing this is the question about how much mathematics students should know and be able to do easily in a first course in physical chemistry and subsequent courses that have physical chemistry as a prerequisite. W
The complete articles and Mathcad documents described in these abstracts are available from JCE Online at http://JChemEd.chem.wisc.edu/JCEWWW/Columns/McadInChem/
W
Like Atkins, I think that we must keep in mind the intellectual quality that distinguishes chemists from other scientists. Chemists typically have strong visual and imaginative skills. This makes it possible for us to imagine the unseen and to create chemical structures and understand them with a minimum of visual cues. This makes chemistry more like abstract art than abstract mathematics. Given this way of thinking, what can we do to give students the mathematical skills and insights needed to better understand the quantitative aspects of chemistry? Here the power of using symbolic equation engines becomes apparent. Mathcad and other symbolic mathematics engines permit concept development by appealing to the visual sensitivity of the chemistry student. With symbolic software, faculty can hide difficult and tedious mathematical processes from beginning students. The software can make the mathematics easier, opening the course for a dramatic infusion of concept building, deeper understanding, critical thinking, and more penetrating insights into physical phenomena through mathematical modeling. In his address in Mobile, Atkins described the Fourier transform as one of the most important mathematical techniques in chemistry. Fourier transforms are central to our major spectroscopic techniques and to the determination of molecular structure by diffraction. Atkins went on to say “the Fourier transforms also give great insight into the nature of physical processes, such as scattering, absorption, and emission. Indeed, it is a rewarding challenge to think about what we would see if we were to look at the world through Fourier transform spectacles. At first sight it would be overwhelmingly different; but as we looked, we would gradually acquire extraordinary insight into the events taking place, and would be able to claim that—now—we understand.” In this column we present the work of three chemists who render the qualitative quantitative in a way that provides a supportive and interactive format for learning a complex topic. Each document or set of documents develops the concepts of the Fourier transform. The documents are similar but different. They will help students to develop a deeper understanding of the Fourier transform and to better appreciate the spectroscopic instruments that are part of the routine practice of chemists. Scott van Bramer sets the stage with a single interactive tutorial through which students can capture the essence of the Fourier transforms and how they lead to recognizable signals recorded in the various spectroscopic experiments. van Bramer’s tutorial document is accompanied by an instructor’s document that can be used for interactive presentations before an entire class. W. Tandy Grubbs provides a different perspective by focusing on the vibrational frequencies of a diatomic molecule. His documents examine harmonic and anharmonic potential functions and extend students skills by solving the equations of motion using the Runge–Kutta method. Mark Iannone also provides an introduction to the Fourier transform in his four documents,
JChemEd.chem.wisc.edu • Vol. 76 No. 2 February 1999 • Journal of Chemical Education
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which overlap with the van Bramer document in some content but not in approach. The style of each author clearly illustrates the rich diversity of application and implementation possible with symbolic mathematics software. The documents presented here provide instructors with an array of tools with which to increase student understanding of the Fourier transform. Each document contains ample student exercises and hands-on activities. The Mathcad documents require Mathcad Plus 6 or higher. They have been tested with Mathcad7. The documents can be obtained from the following JCE Internet address: http:// jchemed.chem.wisc.edu/JCEWWW/Columns/McadInChem/. Acknowledgments We thank the NSF for support of the 1997 NSF-UFE Workshop on Numerical Methods in the Undergraduate Chemistry Curriculum Using the Mathcad Software. Additional support was provided by the NSF’s Division of Undergraduate Education through grant DUE #9455928.
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Journal of Chemical Education • Vol. 76 No. 2 February 1999 • JChemEd.chem.wisc.edu
