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JCE SymMath: Symbolic Mathematics in Chemistry
Theresa Julia Zielinski Monmouth University West Long Branch, NJ 07764-1898
Helping Students Learn Mathematically Intensive Aspects of Chemistry The start of a new year is a good time to start new ventures, and we are pleased to announce that there are big changes in the Mathcad in the Chemistry Curriculum column for 2004. As you can see above, the column’s name has changed to JCE SymMath: Symbolic Mathematics in Chemistry, which reflects a broader scope. As part of the Journal of Chemical Education Digital Library (JCE Dlib) in the National Science Digital Library (NSDL), this column is expanding its content from a collection solely of Mathcad documents to include other symbolic mathematics materials. Our overall goal remains the same: to provide quality materials that support student learning and enhance students’ abilities to understand mathematically intense topics in chemistry. We expect the expanded content of this column to greatly increase the number of users of symbolic mathematics materials and to enhance the effectiveness of JCE and the NSDL in enabling students to learn chemistry. Currently this collection consists of 30 Mathcad documents and five Mathematica notebooks, all of which are available Only@JCE Online (http://www.jce.divched.org/JCEDLib/SymMath/). Also included in the JCE DLib collection are 82 documents from the Symbolic Mathematics for Physical Chemistry Web site (http://bluehawk.monmouth.edu/~tzielins/mathcad/index.htm). Expansion to New Systems We are proceeding to expand the symbolic mathematics column in two ways. First, with support from the NSF through a grant to JCE Dlib, JCE staff are translating many existing Mathcad documents into Mathematica notebooks. Eventually we expect to translate to other systems as well. Second, we encourage anyone who has created materials using Mathcad, Mathematica, and Maple to submit them for possible publication. (The three symbolic mathematics systems listed are those for which we have already identified assistant editors to supervise peer review of materials. We are also looking for anyone who is familiar with another symbolic mathematics system to volunteer to serve as an assistant editor so that we can expand the number of systems included in the collection.)
Introducing Mathematica Notebooks The first set of new documents to be added to the collection consists of Mathematica notebook translations of previously published Mathcad documents (see table on page 156). We are continuing to translate other Mathcad documents to Mathematica. Independent of this translation effort, we are now accepting submissions of original material in Mathematica notebook format for publication in the column and incorporation into the National Science Digital Library. Submissions of Mathematica notebooks should conform to the www.JCE.DivCHED.org
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column guidelines (see below) and be sent to the column editor (
[email protected]) for peer review. Alex Grushow of Rider College is the new assistant editor for Mathematica (
[email protected]).
Maple, Too We are also initiating a collection of Maple-based computer algebra documents. Maple worksheets should conform to the column guidelines (see below) and be sent to the column editor (
[email protected]) for peer review. Louis Kijewski of Monmouth University is the new assistant editor for Maple (
[email protected]). Guidelines for Submitted Documents All submissions should be sent as email attachments to Theresa Julia Zielinski (
[email protected]), the column editor. The guidelines for submission of any document to be considered for inclusion in this column can be found at the JCE SymMath Web site: http:www.jce.divched.org/JCEDLib/ authors/SymMath/. Symbolic mathematics documents for chemical education are most valuable when they expose students to core mathematical techniques and also make it easy for students to use those techniques, thereby enabling them to learn concepts efficiently and effectively. Submissions should include significant pedagogical content, a clear statement of goals, interactive components that support student inquiry, and a carefully constructed bibliography that can lead to further study. Learning Mathematically Intense Chemistry Faculty often complain that students don’t know enough mathematics. While this may be true it is neither the fault of the student nor their mathematics teachers. After all, there is a limit on what can be introduced in a few college mathematics courses and on the time students have to practice and master the mathematics skills that may be required in upper-level chemistry courses. Given that very few of our chemistry majors will go on to pursue careers requiring intensive use of mathematics, we must ask ourselves how we can help students to acquire mathematical skills on a needto-know basis while emphasizing the elegance of the consequences of mathematical methods for understanding chemical concepts. The goal of this column is to disseminate well-crafted instructional materials that support student interactions with mathematically intense chemistry topics. The primary focus is learning chemistry. The mathematics underpinning the chemistry remains important but should not be in the instructional limelight. Rather, mathematics is the flashlight illuminating the subject, not the driving force for studying
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chemistry. Within the framework of our goal, we introduce two new Mathcad documents to provide tools for students to interactively study atomic orbitals and the wave functions of one-dimensional potentials. New Mathcad Releases
The Variation Method and 1-D Potentials In Energies and Wavefunctions for Several One-Dimensional Potentials, Ricardo Metz builds on the detailed explanations of the variational principle presented in the Mathcad particle-in-the-box and harmonic oscillator documents by Grubbs (1) and Dunn (2). Students usually learn about the variational method by practicing it with potentials for which an analytical solution to the Schrödinger equation is known.
While this approach helps students build basic skill with the variational method, it does not show the power of the method or how it can be used to solve problems for which there is no analytical solution to the Schrödinger equation. The document by Metz addresses this issue. For example, Metz permits students to explore the harmonic and Morse potentials simultaneously. Students can see, via overlapping plots, how the wave functions and energy levels for these potentials behave. It is one thing to show a picture of the energy levels of the harmonic oscillator and say they are equally spaced, but it becomes a stronger learning tool when students create these plots and overlap them with the Morse potential energy levels. Through guided inquiry, students discover that the Morse potential permits bond dissociation while the harmonic oscillator does not. The double minimum square potential and
Mathematica Notebooks* dlib
The Carnot Cycle
Introduction to Franck–Condon Factors
by Harold H. Harris, Department of Chemistry, University of Missouri–St. Louis, St. Louis, MO 63121;
[email protected]. Translated by Laura Yindra. Keywords: Thermodynamics, Physical Chemistry, Numerical Methods, Gases Original publication: Harris, H. H. J. Chem. Educ. 2001, 78, 1556.
Variational Methods Applied to the Particle in a Box by W. Tandy Grubbs, Department of Chemistry, Stetson University, DeLand, FL 32720;
[email protected]. Translated by Laura Yindra. Keywords: Computational Chemistry, Quantum Chemistry, Theoretical Chemistry, Numerical Methods Original publication: Grubbs, W. T. J. Chem. Educ. 2001, 78, 1557.
Exploring Light Amplification by Stimulated Emission in Lasers by Michael Waxman, Department of Chemistry, University of Wisconsin–Superior, Superior, WI 54880-2898;
[email protected]. Translated by Laura Yindra. Keywords: Computer Assisted Instruction, Lasers, Physical Chemistry Original publication: Waxman, M. J. Chem. Educ. 2001, 78, 271.
by Theresa Julia Zielinski, Department of Chemistry, Medical Technology, and Physics, Monmouth University, West Long Branch, NJ 07764-1898;
[email protected]. George M. Shalhoub, Department of Chemistry, La Salle University, Philadelphia, PA 19141. Translated by Laura Yindra. Keywords: Physical Chemistry, UV–Vis Spectroscopy, Quantum Chemistry, Laboratory Computing Original publication: Zielinski, T. J.; Shaloub, G. M. J. Chem. Educ. 1998, 75, 1191.
Exploring the Morse Potential by Theresa Julia Zielinski, Department of Chemistry, Medical Technology, and Physics, Monmouth University, West Long Branch, NJ 07764-1898;
[email protected]. Translated by Laura Yindra. Keywords: Physical Chemistry, Internet, Quantum Chemistry, Teaching/Learning Aids Original publication: Zielinski, T. J. J. Chem. Educ. 1998, 75, 1191.
Requirements: Mathematica notebook translations of Mathcad documents require Mathematica version 4.2 or higher; they will run on Windows 95 or higher and Mac OS 8.1 or higher. Document location: http://www.jce.divched.org/ JCEDLib/SymMath/
*These Mathematica Notebooks have been translated from their original Mathcad versions by Laura Yindra of the JCE Staff with support from the NSF through the JCE Digital Library.
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the ammonia inversion potential are also developed in this document. Questions are embedded throughout the document to lead students to discover and analyze wave function and energy level behavior, including the concept of tunneling. This document would be useful for students in introductory quantum chemistry undergraduate and graduate-level courses. The level of work required is not trivial so this document would be best used as a project or group activity with undergraduates in the lecture or laboratory portion of a quantum chemistry course.
Visualizing Atomic Orbitals Orbital Graphing by Mark Ellison provides a welldesigned series of three-dimensional graphics for examining the angular parts of hydrogen-like atomic orbitals. Students are given detailed instructions to create the orbitals and answer questions to promote learning. Although other orbital plot computer algebra documents exist (3, 4), the added value of the guided inquiry approach given by Ellison makes this document especially useful. In the student version the graphs are not included and students are asked to prepare them and learn the concepts interactively. The instructor version gives all graphics and some instructor hints for concepts to bring
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to the attention of students. Instructors can give either document to students depending on the objective of the assignment. In either case student time will be well spent working through this document as homework or as a computer exercise in the inorganic or physical chemistry laboratory. Faculty will also find this a useful tool for lecture and discussion classes in upper-division courses and even for display in general chemistry courses. W
Supplemental Material
Fully interactive computer algebra files, along with static PDF files, are available on JCE Online at http:// www.jce.divched.org/JCEDLib/SymMath/. Literature Cited 1. Grubbs, W. T. J. Chem. Educ. 2001, 78, 1557. 2. Dunn, S. K. J. Chem. Educ. 2002, 79, 1378. 3. Cady, M. P.; Trapp, C. A. A Mathcad Primer for Physical Chemistry; Freeman: New York, 1999. 4. Atkins, P.; de Paula, J. Explorations in Physical Chemistry: A Resource for Users of Mathcad; Freeman: New York, 2002.
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