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JCE SymMath: Symbolic Mathematics in Chemistry
edited by
Theresa Julia Zielinski
Monmouth University West Long Branch, NJ 07764-1898
Teaching Chemistry with Symbolic Mathematics Documents W The typical upper-level chemistry course is packed with information and concepts that instructors feel required to present to students in organized lectures. In physical chemistry one or two semesters is insufficient for exploring the topics in a typical text. Even three semesters would be a stretch for giving students sufficient experience and modern applications. Because of this curriculum crunch, active learning strategies have found little if any place in upper-level courses. Thus, how can one ask teachers to add symbolic mathematics (SM) homework and projects to the mix? How can one expect students to struggle with an SM program while trying to learn a complex set of concepts such as those in physical chemistry? Let’s change orientation and consider that perhaps instructors may be mired in an invalid paradigm. Do we really need to present all the concepts in the textbook in class? Is our role one of being a talking head to students taking notes? Shouldn’t we be extending the skill set of our students by providing them with a strong tool such as an SM computing engine? Given that in many courses the computational tool is a spreadsheet, one could easily substitute an SM engine for lab reports and homework. An SM engine would also be useful for creating templates used during lecture or given to students to help them explore concepts that are impossible to comprehend without extensive number crunching. Student resistance to working with symbolic mathematics software may be the major stumbling block to using SM software, even when the instructor is willing to do so. Many students seem to be permanently joined at a wrist to their calculators. This could be a consequence of using calculators extensively in general chemistry for everything, perhaps even multiplying numbers by ten. Some instructors may also think that students should not be learning about software while taking a course. After all our upper-level courses are not programming courses. What these instructors and students fail to realize is that symbolic mathematics software gives them the power to do calculations that are labor intensive, time consuming, and difficult to do with a calculator. Some might say that one doesn’t need to do number crunching problems to learn the concepts. Students can be given the results to use to support concepts. I argue that the concepts and the computational supports become black boxes to students and that students can come to think that the computations are unimportant or beyond their ability. This attitude certainly does not empower students as problem solvers. Upper-level courses are enriched when there is ample opportunity for students to analyze experimental data using meaningful numerical analysis. Instructor-generated analyses even with careful explanations do not lead to long term learning of concepts. Students need hands-on and minds-on activities with data. Students need to do and can do large calculations that can lead to meaningful learning. The strongest argument for
using symbolic software in upper level-courses is that within SM software, students can wrap their minds around sophisticated algorithms relatively painlessly and only after minimal training. Furthermore within SM software, as in Excel, students can fix data entry or formula entry errors quickly and easily after a bit of investment in learning the fundamentals of the software. With SM software, formerly time consuming, repetitive, and complex calculations are within student reach. When students have success with complex, data-driven problems, they are empowered as learners. There is still more to the importance of symbolic math tools. First, instructors can build on the instruction many students get in their calculus courses. This makes the development of computation skills in math classes very relevant to chemistry courses. Second, the development of critical thinking and logical sequencing of thoughts is strongly supported by use of symbolic software. One cannot develop a working SM document without real understanding of the problem at hand and the orderly presentation of a solution to that problem. Third, when students are required to annotate their SM documents they are engaged in metacognitive processes that further reinforce learning and enhance problem solving skills. This column introduces four new SymMath documents. These documents put special power into the hands of student learners. Students are freed from the difficult code development and given the opportunity to explore topics and learn fundamental concepts. Those students so inclined would also be able to study the foundation algorithms of each document. Constructing Hydrogen Atom Angular Wave Functions and Probability Distributions The method for generating the hydrogen atom angular functions and probability distributions is presented in “Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom” by Kuntzleman, Ellison, and Tippin. In most physical chemistry texts the explicit treatment of each component of the angular functions is not shown. In this document students can explore the Legendre polynomials, the associate Legendre polynomials, the ϕ-dependent function, and the normalization factor for hydrogen atom angular wave functions. Students are led to the stage of drawing the 2p 0 angular function in 3-dimensions. The resulting figures can be rotated and examined by students or used by instructors during class. There are sufficient details in this document for students or instructors to extend the exercises to higher level angular wave functions. The document gives clear exposition of how the angular wave functions and the typical textbook figures are prepared. Two related SM documents are by Ellison (1) in Mathcad and by Kijewski (see below) in Maple formats.
www.JCE.DivCHED.org • Vol. 84 No. 11 November 2007 • Journal of Chemical Education 1885
Information • Textbooks • Media • Resources
Exploring Thermodynamics Using Non-Traditional Systems: Elastomers and DNA In “Exploring Thermodynamics Using Non-Traditional Systems: Elastomers and DNA”, Jeffrey Draves provides an example of how physical chemistry or biophysical chemistry can be enriched with a Mathcad project that studies the elasticity of polymers and extends to stretching DNA using data from the literature. The document provides an interesting introduction using several important physical chemistry concepts. The author includes the central concepts of the first law of thermodynamics, the ideal gas, entropy, equations of state, work, and the Carnot cycle. The mathematical treatment makes use of differentiation, integration, curve fitting, and numerical analysis. Since the document ties together a range of concepts in thermodynamics, it would make an excellent culminating activity for the core thermodynamics sequence of topics. This document would be especially interesting to students of biophysics and biochemistry. Graphing Orbitals in Three Dimensions with Rotatable Density Plots The third SM contribution, “Graphing Orbitals in Three Dimensions with Rotatable Density Plots” by Lou Kijewski, develops the method for preparing probability density plots for complete hydrogen atom wave functions, combining both radial and angular parts. This document, written in Maple, complements that of Kuntzleman et al. described above. In the Kijewski document the method for generating dot diagrams is developed in a way that can be adapted by students to other orbital plots. The author also includes the density plots for some hybrid orbitals. The method for generating a cross section of an orbital is illustrated using the 2s orbital. The same procedure can be used to generate the cross sections of other orbitals.
Since the plots can be rotated, students can examine them and compared them to static figures found in texts. Mastery exercises include suggestions to prepare other plots for wave functions not included in the document. The algorithm for generating the plots is general and extendable to other atomic or molecular wave functions that instructors may wish to develop as teaching resources. Although some of the algorithm components would need to be explained to students, the power of the document is the transparent way the algorithm is developed and how it can be used with students. Mathcad users should also see “Orbital Graphing” by Mark Ellison (1). Exploring Orthonormal Functions The final document in the collection is “Exploring Orthonormal Functions” by Theresa Julia Zielinski. This document was designed as a work sheet for students to use to explore the orthogonality and normality properties of particle-in-a-box wave functions. The document also serves as a preliminary introduction to creating an overlap matrix for a set of functions, namely a set of ten particle-in-a-box wave functions. The resulting matrix is a unit matrix as expected. However, the concept of an overlap matrix developed on this small scale prepares students for the same concept as found in semi-empirical or ab initio calculations. A further application of the particle-in-a-box functions, since they form a complete orthonormal set, is using them as a basis set for curve fitting to other functions. This feature serves as a preparation for students to later examine and use Fourier series expansions. Instructors can use the whole document as is or take parts for specific sections of a quantum chemistry course. Literature Cited 1. Ellison, M. D. J. Chem. Educ. 2004, 81, 158.
1886 Journal of Chemical Education • Vol. 84 No. 11 November 2007 • www.JCE.DivCHED.org
