In the Classroom
W
Pollution Police How To Determine Spectroscopic Selection Rules
Jodye I. Selco* Center for Education and Equity in Mathematics, Science and Technology, California State Polytechnic University–Pomona, Pomona, CA 91768; *
[email protected] Janet Beery Department of Mathematics, University of Redlands, Redlands, CA 92373-0999
How many times have you taught the highly visual yet very abstract spectroscopy portion of your physical or analytical chemistry course, only to be disappointed by students’ generally low level of understanding of optical spectroscopy selection rules and the relationship between symmetry and spectroscopy? In this project students use mathematics and physical chemistry to intuitively yet carefully develop the selection rules for molecular infrared activity, and then use these rules to identify atmospheric pollutants from infrared spectra. Outline of the Project The students, posing as scientists for an environmental testing company, detect and identify atmospheric pollutants such as pesticides, refrigerants, fuels, and products of fuel combustion. Some of these chemicals are released into the atmosphere from factories, cars, or cattle; others evaporate from agricultural fields. Since these chemicals absorb infrared light, they can be detected with an infrared spectrometer. After developing the selection rules for molecular infrared activity and predicting infrared spectra for various suspect molecules, students use these rules to identify atmospheric pollutants from infrared spectra. They then determine whether a given infrared spectrum corresponds to a pollutant from a strawberry field (pesticide), cows, a natural gas pumping station, or an auto repair shop. Students begin by building physical models of an assigned set of molecules, estimating their centers of mass, and then assigning Cartesian coordinate axes using the Mulliken convention. Upon further examination of the physical models, students identify the symmetry elements and operations for each molecule. They then determine the effects of the symmetry operations on the coordinate axes for the molecules
O H
O H
asymmetric stretch
H
O H
H
H
symmetric stretch
bend
Figure 1. Vibrations of water.
and record the results using 1 and 1; in doing so, the students begin to develop their own character tables. After studying the actions of the symmetry operations on the molecular axes, the students examine the vibrations of the molecules using a molecular modeling program, such as Spartan (Figure 1). They determine the actions of the symmetry operations on the normal modes of vibration and compare the results to the transformations of the Cartesian coordinate axes. Since the vibrations that transform in the same way as the coordinate axes are the ones that are infrared-active, the students have just determined the infrared selection rules (Table 1). Although water, which has no degenerate vibrations, is used as an example throughout this assignment, students are also asked to examine the doubly degenerate vibrations of methyl bromide. Since most chemistry majors are not required to take linear algebra, explicit instruction about transformation matrices and how they apply to methyl bromide is provided so that students also can determine the selection rules for degenerate vibrations (Figure 2 and Tables 2 and 3). After determining which vibrations are infrared-active, students use the vibrational energies calculated by the molecular modeling program to generate synthetic infrared spectra.
Table 1. Transformation of the Axes and Vibrations of Water under Symmetry Operations Aspect of H2O Undergoing Transformation
^ E
^ C 2
^ (xz)
^ ( yz)
x Axis
1
1
1
1
y Axis
1
1
1
1
z Axis
1
1
1
1
Bending vibration
1
1
1
1
Symmetric stretch vibration
1
1
1
1
Asymmetric stretch vibration
1
1
1
1
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In the Classroom
Table 2. Axis Transformation Table for CH3Br (Matrices) Axis
^ E
^ C 3
^2 C 3
^
^
^
z
[1]
[1]
[1]
[1]
[1]
[1]
x, y
1 0 0 1
− 12 3 2
−
3 2
− 12
− 12 −
3 2
They then acquire IR spectra of some of the molecules and compare them to the synthetic ones. At the end of the assignment, students are asked to identify a polluter by comparing an infrared spectrum obtained “in the field” to their predicted spectra (Figure 3).
3 2
1
− 12
3 2
− 12
3 2
1 2
−
0 −1
− 12
3 2
−
3 2
1 2
Table 3. Axis Transformation Table for CH3Br (Traces)
Rationale Pollution Police is an interdisciplinary project that has been used in physical and analytical chemistry courses, as well as Abstract Algebra, a junior-level mathematics course. This project could also be used in a graduate-level spectroscopy course. The mathematics underlying modern spectroscopy is group representation and character theory, a field seldom covered in the undergraduate mathematics or chemistry curriculum. Although chemical character tables abound in the literature (1–8), physical chemistry textbooks rarely show how they are constructed, nor do they make clear the connection between symmetry and molecular spectroscopy. This project is intended to show students how quantum mechanical selection rules can be developed. Students see how elementary linear algebra and group theory are used to describe molecular symmetry, how chemical character tables are constructed, and how spectroscopic selection rules are determined from the symmetry. In addition to the connection between symmetry and spectroscopy, students also learn about point groups, symmetry elements, symmetry operations, and degenerate vibrations.
0
Axis
^ E
^ C3
^2 C 3
^
^
^
z
1
1
1
1
1
1
x, y
2
1
1
0
0
0
H
H
Br
C
z axis
H x axis Figure 2. Methyl bromide.
Student Reactions
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80
60
%T
We have found that students who used this project relative to students who did not gained a much deeper understanding of the information contained within chemical character tables and of how symmetry determines the selection rules for light absorption. When asked to determine the selection rules for Raman spectroscopy, students who had undertaken this assignment easily did so without further guidance. In addition, students who had completed this project and had been asked to determine Raman selection rules frequently commented that this project enabled them to see clearly the connection between symmetry and spectroscopy. After doing this assignment, students displayed greater understanding and self-confidence in identifying symmetry elements and symmetry operations, constructing group mul-
40
20
0 4000
3000
2000
Wavenumber / cm
1000
ⴚ1
Figure 3. Infrared spectrum of an atmospheric pollutant.
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In the Classroom
tiplication tables, and determining the symmetries of vibrations and quantum mechanical selection rules. Students better understood degenerate vibrations and the differences between symmetry elements and operations. They felt that this project was an effective and fun way to learn how to assign point groups, how vibrations transform under the actions of the symmetry operations, and how to figure out whether a vibration will be infrared-active. Students also enjoyed using a variety of instructional tools to investigate a current environmental problem.
Upon examination of calculated vibrational energies, students often comment that all of the calculated frequencies are too high by about 10%, but that the exact deviation seems to depend upon the type of vibration. When they examine a variety of basis set inputs or a variety of types of calculations, they also notice that the vibrational frequencies are different, but not necessarily consistently closer to the spectroscopic values. This assignment could be easily adapted so that students compare the frequencies obtained from different levels of calculation and different basis sets with the spectroscopic values. Documentation and Materials Needed The assignment itself as well as the solutions are available as Supplemental Material.W In order to use this project, you will need molecular models, a molecular modeling program such as Spartan, and an infrared spectrometer with a gas sample cell. Most of the infrared spectra are available at http://webbook.nist.gov/chemistry (accessed Dec 2003). Hazards There are no significant hazards for this project unless students are obtaining infrared spectra. Gaseous samples in cylinders are at high pressure. Methane, methyl bromide, allene, ethylene, benzene, and toluene are flammable. Methyl bromide, sulfur trioxide, benzene, and toluene are toxic. Methyl bromide, sulfur trioxide and benzene are suspected carcinogens. Further information about the hazards of these chemicals can be found in the Supplemental MaterialW of this article. Acknowledgments We would like to thank Christopher Brazier for his careful reading of the project and for his helpful suggestions. We also would like to thank Peter Kelly, Teresa Longin, Steven Morics, and Leah O’Brien and their students for testing early versions of this project. Finally, we thank the reviewers for their suggestions.
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Supplemental Material
Notes for the instructor, the assignment itself, assignment solutions, and additional information on associated hazards are available in this issue of JCE Online.
A Warning about Computer Calculations
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This project was originally developed as an Interdisciplinary Lively Application Project (ILAP) with support from an NSF Division of Undergraduate Education grant, DUE9555414, under the aegis of the Consortium for Mathematics and Its Applications (COMAP), Lexington, Massachusetts. For more information about ILAPs and COMAP, visit http:// www.comap.com (accessed Nov 2003).
Literature Cited 1. 2. 3. 4. 5. 6. 7.
Kettle, S. F. A. J. Chem. Educ. 1989, 66, 818–820. Huang, S. Q.; Wang, P. G. J. Chem. Educ. 1990, 67, 34–35. Kettle, S. F. A. J. Chem. Educ. 1999, 76, 675–678. Baraldi, I.; Vanossi, D. J. Chem. Educ. 1997, 74, 806–809. Hardgrove, G. L. J. Chem. Educ. 1997, 74, 797–799. McNaught, I. J. J. Chem. Educ. 1997, 74, 811–812. McQuarrie, Donald A.; Simon, John D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, California, 1997. 8. Cotton, F. Albert. Chemical Applications of Group Theory, 3rd ed.; John Wiley and Sons: New York, 1990.
Further Reading If you or your students wish to learn how group representation and character theory are used in molecular spectroscopy, consult, for instance: 1. Bishop, David M. Group Theory and Chemistry; Dover Publications: Mineola, NY, 1993. 2. Hall, Lowell H. Group Theory and Symmetry in Chemistry; McGraw-Hill: New York, 1969. 3. Herzberg, Gerhard. Molecular Spectra and Molecular Structure: II. Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand: Princeton, NJ, 1945. 4. Wilson, E. Bright; Decius, J. C.; Cross, Paul C. Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra; McGraw-Hill: New York, 1955.
For another ILAP classroom spectroscopy project that makes more explicit use of group representation and character theoretic results, see: 5. Halverson, T.; Varberg, T. Molecular Vibrations and Symmetry; Macalester College: St. Paul, MN, 1997.
If you or your students wish to see examples of environmental spectroscopic experiments, consult, for instance: 6. Allen, H. C.; Brauers, T.; Finlayson-Pitts, B. J. J. Chem. Educ. 1997, 74, 1459–1462. 7. Elrod, M. J. J. Chem. Educ. 1999, 76, 1702–1705.
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