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Making a Sundial
Before there were mechanical clocks, people used the Sun to tell time. As Earth rotates about its axis, the Sun appears to be moving through the sky, changing its position-- even though it is really we who are spinning. Sundials use the change in the Sun's position to measure time. In the morning, your shadow points long into the west as the sun rises in the east. As noon approaches, your shadow gets shorter. Then your shadow points eastward, increasing again, as the Sun sets.
The length of your shadow also depends on your longitude (distance east or west of the prime meridian, measured by imaginary lines running from the North Pole to the South Pole). In northern winters, the Sun is low in the southern sky and casts a long shadow through most of the day. This results from the tilt in Earth's axis, which gives us seasons.
- sheet of large paper
- 3-to-4-mm-thick sheet of corrugated cardboard or plywood
- long nail
1. Find a sunny location on a clear day. Begin early in the morning.
2. Tape a sheet of paper to the cardboard.
3. Hammer a nail into the center of the paper, just far enough in so that it stands firmly on its own.
4. Mark the shadow of the nail at 1-hour intervals (e. g., 8: 00 A. M., 9: 00 A. M., etc.). Mark the shadow of the head of the nail, then draw the shadow of the nail using a straightedge or ruler. Mark the time of day. It is very important that you do not change the position of your board during the day.
5. At the end of the day, you will have a functional sundial! Record the date and location (city, state) on your sundial.
6. You can use your paper as a template to make a sundial out of more durable and attractive materials, such as wood, plastic, metal, and so on.
7. If you move your sundial, recalibrate it by positioning it to read the proper time. You'll have to recalibrate it twice a year anyway if your region goes on and off of daylight saving time.
8. As an extension to this activity, you can make sundials at different times during the year and compare them. The position of the Sun in the sky and hence the shadow changes with the rotation and revolution of Earth.0471358339.txt |
The premise on which this text is based is that the vast majority of chemical phenomena may be qualitatively understood by the judicious use of simple orbital interaction diagrams. The material borrows heavily from the pioneering work of Fukui [1, 2], Woodward and Hoffmann , Klopman , Salem , Hoffmann , and many others whose work will be acknowledged throughout including Fleming: Frontier Orbitals and Organic Chemical Reactions , from which a number of illustrative examples are extracted. If there is uniqueness to the present approach, it lies in the introduction of the x and ß of simple Hückel molecular orbital theory as reference energy and energy scale on which to draw the interaction diagrams, mixing o and o* orbitals and nonbonded orbitals with the usual p orbitals of SHMO theory on the same energy scale. This approach is diffcult to justify theoretically, but it provides a platform on which the reader can construct his or her interaction diagrams and is very useful in practice. Numerous illustrations from the recent literature are provided.
The book is intended for students of organic chemistry at the senior undergraduate and postgraduate levels and for chemists in general seeking qualitative understanding of the (often) quantitative data produced by modern computational chemists . All reactions of organic compounds are treated within the framework of generalized Lewis acid--Lewis base theory, their reactivity being governed by the characteristics of the frontier orbitals of the two reactants. All compounds have occupied molecular orbitals and so can donate electrons, that is, act as bases in the Lewis sense. All compounds have empty molecular orbitals and so can accept electrons, that is, act as acids in the Lewis sense. The ``basicity'' of a compound depends on its ability to donate the electron pair. This depends on the energy of the electrons, the distribution of the electrons (shape of the molecular orbital), and also on the ability of the substrate to receive the electrons (on the shape and energy of its empty orbital). The basicity of a compound toward different substrates will be different, hence a distinction between Lowry--Bronsted basicity and nucleophilicity. A parallel definition applies for the ``acidity'' of the compound. The structures of compounds are determined by the energetics of the occupied orbitals. Fine distinctions, such as conformational preferences, can be made on the basis of maximization of attractive interactions and/or minimization of repulsive interactions between the frontier localized group orbitals of a compound. All aspects are examined from the point of view of orbital interaction diagrams from which gross features of reactivity and structure flow naturally. The approach is qualitatively different from and simpler than, a number of alternative approaches, such as the VBCM (valence bond configuration mixing) model  and OCAMS (orbital correlation analysis using maximum symmetry) approach [10, 11].
The organization of the text follows a logical pedagogical sequence. The first chapter is not primarily about ``orbitals'' at all but introduces (or recalls) to the reader elements of symmetry and stereochemical relationships among molecules and among groups within a molecule. Many of the reactions of organic chemistry follow stereochemically well-defined paths, dictated, it will be argued, by the interactions of the frontier orbitals. The conceptual leap to orbitals as objects anchored to the molecular framework which have well-defined spatial relationships to each other is easier to make as a consequence. Whether or not orbitals interact can often be decided on grounds of symmetry. The chapter concludes with the examination of the symmetry properties of a few orbitals which are familiar to the student.
The second chapter introduces the student to ``orbitals'' proper and offers a simplified rationalization for why orbital interaction theory may be expected to work. It does so by means of a qualitative discussion of Hartree--Fock theory. A detailed derivation of Hartree--Fock theory making only the simplifying concession that all wave functions are real is provided in Appendix A. Some connection is made to the results of ab initio quantum chemical calculations. Postgraduate students can benefit from carrying out a project based on such calculations on a system related to their own research interests. A few exercises are provided to direct the student. For the purpose of undergraduate instruction, this chapter and Appendix A may be skipped, and the essential arguments and conclusions are provided to the students in a single lecture as the introduction to Chapter 3.
Orbital interaction theory proper is introduced in Chapter 3. The independent electron ( Hückel) approximation is invoked and the effective one-electron Schroüdinger equation is solved for the two-orbital case. The solutions provide the basis for the orbital interaction diagram. The effect of overlap and energy separation on the energies and polarizations of the resulting molecular orbitals are explicitly demonstrated. The consequences of zero to four electrons are examined and applications are hinted at. Group orbitals are provided as building blocks from which the student may begin to assemble more complex orbital systems.Chapter 4 provides a brief interlude in the theoretical derivations by examining specific applications of the two-orbital interaction diagrams to the description of o bonds and their reactions.
In Chapter 5, conventional simple Hückel molecular orbital (SHMO) theory is introduced. The Hückel x is suggested as a reference energy, and use of |ß| as a unit of energy is advocated. Parameters for heteroatoms and hybridized orbitals are given. An interactive computer program, SHMO, which uses the conventions introduced in this chapter, is available on the Web .
Chapters 6--11 describe applications of orbital interaction theory to various chemical systems in order to show how familiar concepts such as acid and base strengths, nucleophilicity and electrophilicity, stabilization and destabilization, and thermodynamic stability and chemical reactivity may be understood.
Pericyclic reactions are described in Chapter 12 as a special case of frontier orbital interactions, that is, following Fukui . However, the stereochemical nomenclature suprafacial and antarafacial and the very useful general component analysis of Woodward and Hoffmann  are also introduced here.
The bonding in organometallic compounds between the metal and C and H atoms is briefly described in Chapter 13.
Chapter 14 deals with orbital correlation diagrams following Woodward and Hoffmann . State wave functions and properties of electronic states are deduced from the orbital picture, and rules for state correlation diagrams are reviewed, as a prelude to an introduction to the field of organic photochemistry in Chapter 15.
In Chapter 15, the state correlation diagram approach of the previous chapter is applied to a brief discussion of photochemistry in the manner of Dauben, Salem, and Turro . A more comprehensive approach to this subject may be found in the text by Michl and Bonacic-Koutecky , Turro , or Gilbert and Baggott .
Sample problems and quizzes, grouped approximately by chapter, are presented in Appendix B. Many are based on examples from the recent literature and references are provided. Detailed answers are worked out for many of the problems. These serve as further examples to the reader of the application of the principles of orbital interaction theory.