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JCE SymMath: Symbolic Mathematics in Chemistry
Theresa Julia Zielinski Monmouth University West Long Branch, NJ 07764-1898
Learning That Prepares for More Learning: Symbolic Mathematics in Physical Chemistry Recently, during a PBS program about the history of education in New Jersey, the speaker commented that education is not preparation for a job; rather education is preparation for more education. I would paraphrase this by saying that learning chemistry is not about knowing chemistry; rather it is preparation for learning more chemistry. This is especially true in physical chemistry where students need excellent tools to enhance the potential for continued learning within the discipline and across the other subdisciplines of chemistry. The various symbolic mathematics programs are ideal tools for the chemistry student at any stage of sophistication. Beginners can work with well-crafted templates to learn new concepts while more advanced learners can increase their understanding of theoretical models and manipulate large sets of experimental data. In this issue of the symbolic mathematics column five Mathcad templates focus student learning on pressure-volume work, the Boltzmann distribution, the Gibbs free energy function, intermolecular potentials and the second virial coefficient, and quantum mechanical tunneling.
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Hot or Cold Nature of Negative Absolute Temperatures In a third template, Temperature As a Measure of the Distribution of Particles over Energy States, Ferguson uses guided inquiry to explore the Boltzmann distribution and evaluate the number distribution of a set of particles over four energy states as a function of temperatures. This template, by providing all required symbolic expressions and guided inquiry activities, effectively leads students to the idea of a temperature inversion. The strength of Ferguson’s templates lies in the clear instructions and guiding questions that help students who are learning thermodynamic concepts but who lack extensive experience with Mathcad. Real Gases and the Second Virial Coefficient
In Work Done during Reversible and Irreversible Isothermal Expansion of an Ideal Gas, Arthur Ferguson guides students through a sequence of activities leading to the discovery of how an increasing number of isothermal expansion steps of an ideal gas yields the work for the corresponding irreversible path between two states. When the series is extended to a very large number of steps an approximation of the reversible isothermal expansion work is obtained that can be checked by integration. In the template students are given the required equations and guided inquiry questions. This focuses student attention on the work concept rather than the process of debugging symbolic code. The guided inquiry questions promote critical thinking and strengthen understanding of the reversible work concept.
In Intermolecular Potentials and the Second Virial Coefficient document Patrick Holt draws the student’s attention to the second virial equation of state and the relationship of that equation of state to potential functions commonly used by chemists in molecular mechanics calculations, namely the hard sphere model, the square well potential and the Lennard-Jones potential. Students use each potential to compute the second virial coefficient for several materials and the temperature dependence of the virial coefficient. They then relate the virial coefficient to compressibility and molecular properties. Bold and highlighted regions in gray show the instructor’s notes and solutions to selected exercises that should be removed in the versions used by students. Equations for students to use are provided throughout the document. Holt also provides the derivation of the second virial coefficient using the Lennard-Jones potential. After using this document, students should have a firm understanding of three important potential functions and the limitations of these functions.
Gibbs Free Energy and Equilibrium
Tunneling through Barriers
In Gibbs Free Energy of a Chemical Reaction System As a Function of the Extent of Reaction and the Prediction of Spontaneity, Ferguson uses the guided inquiry approach to lead students to a fuller understanding of the Gibbs free energy for a chemical reaction and the relationship of that free energy with the extent of a reaction and its spontaneity. By working with the template students are expected to learn the shape of the free energy versus extent of reaction plot, how ∆rxnG affects the shape of the plot, and the significance of the plot with respect to the spontaneity of the reaction. With this template students can achieve a moderately sophisticated understanding of the free energy concept and chemical reaction spontaneity.
In the template Potential Barriers and Tunneling, Mark Ellison examines one of the fundamental concepts in introductory quantum chemistry. By providing the required mathematical expressions, Ellison enables students to focus on the interpretation of the wave function in the various regions of the barrier problem. Mathcad plots facilitate the guided inquiry activities for students. Extension of the exercise to scanning tunneling microscopy is a major strength of this template that lets students immediately see the experimental importance of tunneling. The document concludes with an exploration of the potential significance of tunneling in chemical reactions. The progression from theory to microscopy to chemical reactions brings several levels of significance to stu-
Isothermal Gas Expansions and Work
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Vol. 81 No. 4 April 2004
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Journal of Chemical Education
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dents who often question why one would bother to think about tunneling in the first place. In the Curriculum This collection of five symbolic mathematics templates brings students from basic thermodynamics concepts through to real equations of state and potential functions for real gases up to the use of potential functions as applied to tunneling barriers in quantum mechanics. These documents form a good basic core for any instructor using symbolic mathematics
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in physical chemistry courses. The documents all require Mathcad8 or higher. PDF files are available for those wishing to adapt these templates to other symbolic mathematics programs. W
Supplemental Material
Fully interactive computer algebra files along with static PDF files are available on the JCE Online Web site at http:// www.jce.divched.org/JCEDLib/SymMath/.
Vol. 81 No. 4 April 2004
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www.JCE.DivCHED.org
