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JCE SymMath: Symbolic Mathematics in Chemistry
Theresa Julia Zielinski Monmouth University West Long Branch, NJ 07764-1898
A SymMath Good Neighbor Policy Those of us who use symbolic mathematics when teaching various chemistry courses constitute a community that can create and maintain a rich reservoir of teaching materials. Through the efforts of this community the SymMath collection currently is a repository of over 100 individual symbolic mathematics documents, most in Mathcad format, but also some in Mathematica and Maple. The collection consists of work contributed by more than 35 different teaching colleagues from about 25 different teaching locations. A substantial number of users, instructors, and students have downloaded materials from the collection. At JCE the SymMath collection has an average of about 339 visits and 240 downloads per day. Knowing this prompts the consideration of a good neighbor policy through which the community can nurture and support new and current users while building the collection. The Nature of a Community A digital community is much like a regular community, namely a group having common interests. The interests of the SymMath community include developing, maintaining, and disseminating high quality, mathematically rich digital instructional resources for teaching chemistry. The community focuses on using Symbolic Mathematics Engines (SME) such as Mathcad, Mathematica, and Maple for science instruction and for facilitating student use of these engines as a means to promote learning topics efficiently and deeply. The community also supports its members through peer review of SME documents and publication of these documents in JCE as part of the SymMath collection. However, there are many silent members of our community: those of us who are using SME software in our classrooms yet are not active contributors to the SME collection and the peer review process. If you, as a more reticent colleague, would contribute your expertise and digital SME documents to the collection, the entire community would benefit. Supporting the Community There are several things that support our community. First, the current SymMath collection files are easy to access and read for anyone using any SME software. The URL for the JCE peer reviewed subset of the SymMath collection is http://www.jce.divched.org/JCEDLib/SymMath/index.html. Links to the JCE peer reviewed documents and the full native open access open review documents are at http:// bluehawk.monmouth.edu/~tzielins/mathcad/index.htm. This year we will be moving the open-access, open-review documents to JCE where they will remain available to all members of the community, whether or not they subscribe to the JCE. All documents in the collection will be identified as either full learning objects, documents that are complete, standalone expositions of a topic in chemistry, or as digital assets, documents that contain a kernel of information that can be combined with other assets to form a new learning object or 1724
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added to existing learning objects to expand their usefulness. The availability of PDF files of all contributed documents also makes the content of any document in any SME format available to any user of the collection. A second support for the community is having components and documents of the collection that are short, easy to use, based on the most fundamental processes of an SME engine, and address the earliest topics that appear in a course. Such documents help students learn to use the SME in context of the subject being discussed in class. From that point on the student can be led to become proficient in using the SME for a variety of laboratory reports and homework assignments. The SME collection already contains many files of this type and community members can help others by contributing to the open-review collection the introductory SME documents that they use to help students learn the software in the context of the discipline. Making available the same materials in multiple formats is a third way to support the SME user community. Currently there are only a few examples of translations of Mathcad documents into Mathematica and Maple. More translations are needed and it is important to note that a translation generates a published abstract in the print Journal. The key to translation is to make the document as nearly identical to the original as possible within the parameters of the alternative SME engine. Being an SME Good Neighbor As a colleague in the SME community, there are several things that you can do to be a good neighbor. These include • Contributing SME documents for inclusion in the open review or peer review sections of the collection. Submissions should be sent electronically to the SymMath feature column editor,
[email protected]. • Providing a short abstract for each submission and formatting your SME document to include the major components described in the author instructions at http:// www.jce.divched.org/JCEDLib/SymMath/authors/index.html. • Volunteer to be a SymMath (and JCE) Peer Reviewer. Completing a JCE reviewer profile form, available at http:// www.jce.divched.org/Contributors/Reviewers/index.html, will get you into the JCE reviewer data base. Be sure to check the Mathematics/Symbolic Mathematics item in the Topics section of this form. • Contribute to the community by sharing tacit knowledge, informal knowledge about how things are done in the classroom or with the SME tool. Such knowledge is usually difficult to collect and organize for sharing with the community. One simple example is the question asked by a colleague with excellent Mathcad skills, “how does one remove a page break from a Mathcad document?” A second broader question was, “Where do I find a clear exposition of programming within Mathcad?”. Questions like
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Information these, along with responses, will be collected and shared though a JCE Discussion Forum.
In support of the SymMath community we present five new documents for your consideration, four in Mathcad and one in Maple. If you use these, please add your observations to the SymMath forum and consider contributing an SME document to the collection. Symbolic Mathematics and the H Atom The template, Symbolic Math Approach To Solve Particle-in-the-Box and H-Atom Problems, by Todd Hamilton, guides junior and senior chemistry students through the symbolic mathematics derivation for the solution to the H-atom Schrödinger equation for the 1s orbital. Students’ practice with the particle-in-the-box gives them experience with the basic symbolic mathematics steps that they then apply to the H-atom. This document is an excellent way for students to learn to use the symbolic processor of Mathcad in a guided inquiry manner. Finding Molecular Vibrational Frequencies from HCl to SO2 In Finding Molecular Vibrational Frequencies from HCl to SO2, Chen and Zielinski direct users along a guided inquiry study of normal mode analysis for several small molecules. The study starts with an examination of classical equations and writing quadratic forms for the potential and kinetic energy of oscillators. Matrices representing the quadratic forms are prepared and then the eigenvalues and eigenvectors are obtained with Mathcad standard functions. The eigenvalues are normal mode frequencies and the eigenvectors are the displacements of the oscillators. Included in the document is a discussion of Hermitian matrices and proofs of some properties of these matrices. The matrix method developed in the document is applied to HCl, CO2, and SO2. A significant aspect of the document is the direct way that the eigenvectors are represented pictorially for students to study and even draw themselves. Seventeen exercises help users to practice the various concepts introduced in the document. This document would be suitable to use with graduate students and advanced undergraduate students in both chemistry and physics courses. Following the Extent of a Reaction with the Help of Maple In Following the Extent of a Reaction with the Help of Maple, Suárez and Coto outline the development of the mathematical treatment of ⌬rG as a function of the extent of a reaction, . The procedure is applied to the ammonia reaction, 1/2 N2 ⫹ 3/2 H2 ⫽ NH3. Using the provided code students can supply the data needed to complete the same study for other reactions. For the ammonia reaction students are requested to analyze the behavior of the reaction with respect to the extent of the reaction and then asked to prepare plots for the reaction at different pressures and temperatures. Finally, students are given ample opportunity to practice the concepts by preparing a companion document for the formation reaction of water. The solution to the ammonia exercise can be removed from documents given to students. For www.JCE.DivCHED.org
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Maple users this is a very useful teaching tool because it has sufficient detail to enable students to use it successfully and to prepare similar documents for other reactions. A Summary of Statistical Thermodynamic Calculations In the Summary of Statistical Thermodynamics Calculations, Zielinski and Young illustrate the fundamental equations for calculating the thermodynamic properties of molecules and equilibrium constants. The document builds students’ skill with partition functions and leads students to successful use of these functions to determine heat capacities, Gibbs and Helmholtz free energies. Computing the equilibrium constant of the dissociation of N2 illustrates the statistical thermodynamic calculations. Later, the heat capacities of H2O and CO2 are compared in order to foster an understanding of the significance of linear versus bent molecules and their heat capacities. The effect of a low-lying electronic energy level on the heat capacity of NO is also examined. After completing this Mathcad document, students should be able to compute the thermodynamic properties for molecules and reactions that interest them. This document is suitable for use in typical physical chemistry courses where the time for instruction on statistical thermodynamics may be limited. The original form of this document was written in 1997 with close collaboration between the authors. This newer version was updated by Zielinski in 2006. Sid Young passed away in July 2004. The publication of this new version of the document is dedicated to the memory of Sid Young and Joseph Noggle, who were pioneers in the use of SME software to aid physical chemistry instruction. The Diffusion Game—Using Symbolic Mathematics Software To Play the Game on a Large Scale In Diffusion Game, Tandy Grubbs presents a clear and systematic exposition of diffusion through the mechanism of using dice to determine the location of a particle on the left or right side of a two-chamber container. Through this students learn about the approach to equilibrium by diffusion. The document then goes on to discuss statistical entropy, the Stirling approximation, and the application of these to the two-chamber system. A plot of S(T ) shows how the entropy approaches a maximum and that deviations from the maximum beyond equilibrium are relatively small. The entropy calculations lead directly into a study of Fick’s law of diffusion. A further extension takes the student to a study of a two-level reaction equilibrium. The document concludes with an introduction to autocatalytic functions and the Onsager regression hypotheses that can be assigned to advanced students. Each component of the document includes instruction on Mathcad syntax and student exercises to build understanding. This document would be a good exercise for physical chemistry students, especially as one reviewer noted, there never seems to be sufficient time to do statistical thermodynamics in a satisfactory way. With this document students will at least get some idea of how ensembles model chemical– physical systems, even if they are not called by that name in the document. Typical students will require instructor assistance to complete the latter fourth of the document successfully.
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