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, span the chemistry curriculum and are frequent topics of discussion among faculty. We can try to span the extreems by 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 mini-treatment 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-to-noise 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 the Signals and Noise Mathcad document described here. Students in our instrumental analysis course were struggling with understanding the two-paragraph definition found for signal-to-noise in their textbook. The students successfully learned the concept through the material presented in the Signals and Noise document described below.

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-to-noise 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 and given the successive conditions that one of the mirrors is 100%, 90%, and 80% reflective. The document then proceeds to guide 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.