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Mathcad in the Chemistry Curriculum
Theresa Julia Zielinski Monmouth University West Long Branch, NJ 07764-1898
Using Symbolic Software to Facilitate Learning When teaching complex chemical concepts we can use a range of techniques. Two extremes along the traditional instructional spectrum are to fill the blackboard with diagrams and equations accompanied by verbal explanations or to give brief summaries and permit students to treat the concept as a black box through which they churn numbers. These two extremes of the teaching conundrum, balancing the breadth of concepts covered in a course with depth of understanding of those concepts, recur throughout the chemistry curriculum and are frequent topics of discussion among faculty. One approach to spanning these extremes is the appropriate use of symbolic mathematics software. In this edition of the Mathcad column I present two documents that illustrate two approaches to depth and breadth for the instruction of difficult topics in the analytical chemistry or physical chemistry portion of the curriculum. In one we see a depth that many instructors will appreciate. In the other we see a carefully constructed minitreatment of topic framed within the boundaries of the limits of the undergraduate physical chemistry curriculum. The first document, “Exploring Digital Signals and Noise in Instrumental Analysis”, is designed to allow students to obtain in-depth experience with the concept of signal-tonoise and recognize the advantages of ensemble averaging and digital filtering of analytical signals. These traditional junior- or senior-level Instrumental Analysis topics present challenges for teaching by instructors and to learning by students. My own recent experience with this topic came at the same time that I received this Mathcad document. Students in our instrumental analysis course were struggling with understanding the two-paragraph definition found for signal-to-noise in their textbook. They successfully learned the concept through the material presented in the document. The signals and noise document provides an instructional
depth that will allow students to determine the signal-to-noise ratio of an analytical signal, explain the relationship between ensemble averaging and signal-to-noise ratio, describe the effects of analog and digital filtering on an analytical signal, explain the relationship between filter bandwidth, signal-tonoise ratio, and signal smoothing, and finally, explain the benefits of combining ensemble averaging with digital filtering of an analytical signal. The document provides clear illustrative examples with well-designed exercises that illustrate the techniques of averaging and filtering noise. In the “Exploring Light Amplification by Stimulated Emission in Lasers” document we find a different approach. This compact document was designed to help students explore a single concept modeled by a single equation, the equation for the light output of a laser. Instructors who want to provide students with a basic understanding of how the intensity of a laser beam depends on the length of the laser cavity, the active medium, and the reflectivity of the cavity mirrors will find this document very useful. To use the document effectively students should know the basis of lasing action. The laser exploration prepares the students for further work by having them estimate the intensity of a laser beam output after 100 ns, given the successive conditions that one of the mirrors is 100%, 90%, and 80% reflective. The document then guides students to the mathematical expression for the output laser intensity as a function of time. Students next explore the affect of various parameters on the output laser light intensity. There are ample exercises imbedded in the document and also opportunities for writing explanations of observations on the calculations. The document would also be useful as a classroom instructional tool followed by practice as homework. A short essay for instructors accompanies the document.
Mathcad in the Chemistry Curriculum is an online feature of JCE Internet. The complete Mathcad documents described here can be obtained online from
http://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/ Access to the working Mathcad versions of these documents is restricted to JCE subscribers. Non-interactive representations of the Mathcad documents are also available as Adobe Acrobat PDF files.
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Journal of Chemical Education • Vol. 78 No. 2 February 2001 • JChemEd.chem.wisc.edu
Information • Textbooks • Media • Resources
Exploring Digital Signals and Noise W in Instrumental Analysis: SignalsNoise.mcd Augustus W. Fountain III, Department of Chemistry and Photonics Research Center, United States Military Academy, West Point, NY 10996;
[email protected] The fundamental concepts of noise and signal-to-noise ratio are central to the discipline of analytical chemistry. Since chemical instrumentation dominates the modern laboratory, it is critical that students understand the role noise plays in limiting the precision of a measurement. It is equally important that they understand how to implement active and passive means to mitigate noise. The purpose of this Mathcad document is to allow the student to gain a familiarity with the concepts of signal-to-noise ratios and to explore the advantages of ensemble averaging and digital filtering analytical signals. Simulated noisy signals are used to guide the student through a series of individual exercises (Fig. 1). These exercises culminate in a capstone experience in which they simultaneously apply all of the concepts. Exploring Light Amplification by Stimulated Emission in Lasers: Laser.mcd and Notes.doc
Figure 1. Ensemble averaged signal and original signal.
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Michael Waxman, Department of Chemistry, University of Wisconsin-Superior, Superior, WI 54880-2898
This document allows the students to use a simple approach to study the time evolution of the power output of a laser and its dependence upon the amplification coefficient, the length of the active medium, and the reflectivity of the cavity mirrors (Fig. 2). The document starts with a problem on finding the laser power at a given time for the different values of the mirror reflectivity. After solving this problem, the students are guided to predict the optimal value of the mirror reflectivity using simple calculus, to check their prediction using Mathcad, and to present a physical explanation of their findings. Then they use Mathcad to analyze the dependence of the output intensity on the amplification co-
Figure 2. Output laser signal as a function of the fraction of light intensity that passes through one of the laser mirrors.
efficient, the length of the cavity, and time passed since the laser was switched on. Afterwards, students design a hypothetical laser and present their design strategies and the results in their report. Finally, we discuss the limitations of the simple model used and outline the more rigorous approach to study the dynamics of laser generation. The material in this exercise can be used within a first course in Physical Chemistry where lasers and their applications in chemistry are introduced. Instructor notes for this document give the rationale for the model used and suggestions for completing some exercises.
JChemEd.chem.wisc.edu • Vol. 78 No. 2 February 2001 • Journal of Chemical Education
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