Simplifying Difficult Mathematical Concepts in Chemistry Courses Russell W. Maatman Dordt College, Sioux Center, IA51250 Usually physical chemistry students must accept on faith the statement that quantum numbers arise naturally in the process of solving the Schrodinger equation for the hydrogen atom problem. I t seems that the Schrodinger equation is too diff~cultto use. Most students are willing to make this leap of faith from the equation to auantum nums are a bers. I.:ven 60, in every physical rheniistry c l a ~ there few student;; who feel unsatisfied with this lt!np-of-faithintroduction of quantum numbers, which, afteiall, play a n extremely important role in every chemistry course. If the very students who find this leap-of-faith situation unsatisfying were able to surmount this first barrier in the use of quantum mechanics, they would have the potential to become productive in this area later. Perhaps the same situation exists in other parts of physical chemistry, such as the theoretical development of rate laws for chemical reactions. Therefore, I have written two syllabi, one on using the time-independent Schrodinger equation for the hydrogen atom. so that the three spatial auantum numbers areWproduced,and the other o n the formulation of rate laws. The purpose of this note is to describe these syllabi and offer them to anyone who might be able to use them. Both svllabi differ from the usual svllabi we use for our classes in four ways. .Almost all the steps from starting point to conclusion are given. The very few unexplained leaps are identified. The mathematical steps are very small. The goal is that the students will be able to see, without writing down anything, how each step fallows from the previous one. T h e development from starting point to result avoids describing nonessential, and therefore distracting side issues, even though they may be interesting. The level of mathematics required is kept to a minimum. For example, a differential equations course is not a prerequisite for using the Schrodinger equation to develop quantum numbers (seebelow). Although the various parts of the development in each syllabus are found in the literature, the entire picture is probably inaccessible to students. Furthermore, to help unify these parts, a single symbolic system is used in each syllabus. I n the syllabus, "Origin of the Spatial Quantum Numbers n, 1, and ml," the student need not be familiar with any mathematics beyond differential calculus. The syllabus begins with a derivation of the time-independent Schrodinger equation in spherical coordinates. I t is explained that the key part of this procedure, which depends upon an analogy, is not a true derivation. The resulting differential equation is separated into three differential equations; in each, the variable is a function of only one of the three spherical coordinates. To solve the time-inde-
pendent Schrodinger equation, one must solve these three equations for the three functions. In this syllabus, the complete solutions are not obtained. But the syllabus does show how the process of solvine the equations produces the three spatial quantum numbeis and their allowed values. I n one step involving the solution of one of the three equations, knowledge of more than differential calculus is required; and so to make this step easier to understand, a plausible alternate procedure is presented. The other two differential equations are solved after the functions have been expressed in terms of power series. The power-series method is, of course, included in differential equations courses. But the method requires no more than the familiar idea (often taught in algebra eourses)that some functions can be expressed in power series and the ability to differentiate functions.
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The syllabus, "Introduction to the Rate Laws of Chemical Reactions," consists, given such-and-such a reaction mechanism. of the mathematical development of rate rate laws and experimentai results is not laws. taken up. The development is initially from first principles-e.g., the assum$ion that gas moiecules can coilidto the simplest rate laws. The first part of the syllabus is suitable for general chemistry students. The remainder is for physical chemistry students. The reason for this syllabus is that the concept of a rate law is foreign to most, perhaps all, of the other chemical ideas of first courses in chemistry. Furthermore, even when students do comprehend the rate law concept, they still might have difficulty in understanding how some rate laws can be so com~licated, involving, for example, fractional exponents and factors in the denominator. Considerable attention is given to rules for developing rate laws. Noncatalyzed and catalyzed (both homogeneous and heterogeneous) reactions are considered. For some of the rate laws, real examples are given. The syllabus proceeds from simple to complicated cases. They include forward and reversible gas-phase reactions; reactions consisting of anywhere from two to five steps, with various combinations of slow and rapid steps and reversibility: .. chain reactions; and pseudo-first-orde;gas reactions. These syllabi have been used with the physical chemistry students of Pamela Veltkamp a t Dordt College; in addition, the Schrodinger equation syllabus bas been distributed to many other physical chemistry instructors. Both syllabi are available from the author a t no cost.
ela at in^
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Acknowledgment Suggestions made by Pamela Veltkamp and her physical chemistry students a t Dordt College have been very helpful.
Volume 72 Number 12 December 1995
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